On the projective geometry of curves of

genus one, and an algorithm for the jacobian of such a curve

Alexander R. Perlis

A Dissertation Submitted to the Faculty of the

Department of Mathematics

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

In the Graduate College

The University of Arizona

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Copyright \begin{matrix}c \\ ◯ \end{matrix} 2004 Alexander R. Perlis. All rights reserved.

Document composed and printed using the L^{A}TEX document preparation system.

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2

Statement by Author

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be obtained from the author (who is the copyright holder).

Acknowledgments

The idea of the algorithm was presented by Minhyong Kim in a series of informal lectures when we were working on [AKM^{+}01]. Much of the material in this dissertation was worked out in extensive conversations with Minhyong Kim and William McCallum, both of whom I owe a great debt of gratitude.

Both throughout this dissertation and in other material related to [AKM^{+}01], there are contributions from conversations with Greg Anderson, John Brillhart, John Cremona, Larry Grove, William Hoffman, Hermann Karcher, Kirti Joshi, Benjamin Levitt, Klaus Lux, David Marshall, Susan Marshall, Barry Mazur, Jorge Morales, Catherine O’Neil, Robert Perlis, Christopher Rasmussen, Nicholas Shepherd-Barron, Dinesh Thakur, and Douglas Ulmer, among others—my sincere apologies to anyone whose name I have forgotten to mention. I thank everyone for their time, encouragement, and words of wisdom.

Abstract

Given equations with k-rational coefficients that define a curve C of genus 1 over a perfect field k, can we find equations that define its jacobian J_{C}? The problem is trivial when the degree n of a k-rational divisor on C is equal to 1. For the cases 2 ≤ n ≤ 4, certain standard forms for C appear classically, and the classical invariant theory of those forms turns out to contain equations that define J_{C}. This modern interpretation of classical results was explained for n = 2 in 1954, for n = 3 in 2001, and for n = 4 in 1996. A standard form for C and its invariant theory was worked out by Tom Fisher for n = 5 in 2003, again leading to equations for J_{C}.

In the present work, the problem is solved algorithmically for all n ≥ 3. (As in the classical approach, we must assume the characteristic of k does not divide n.) The basic idea, given to us by Minhyong Kim, is to embed C in P_{k}^{n−1} using the divisor of degree n, then to explicitly describe as matrices the finite Heisenberg group that corresponds to the n-torsion J_{C}[n] on the jacobian, and then to determine equations for the quotient of C by the Heisenberg group, giving us the sought jacobian: C/J_{C}[n] ≅ J_{C}. The Heisenberg matrices also allow us to compute the points of hyperosculation on C, which is a k-rational orbit under the action of J_{C}[n] and thus gives the origin for the group law on J_{C}. Our algorithm relies on techniques from the theory of Gröbner bases, and on techniques from the invariant theory of finite groups.

In presenting the background material to our algorithm, we develop the theory of curves of genus 1 with an attached k-rational divisor class, and the theory of non-degenerate degree n curves in P_{k}^{n−1} of genus 1. We thus state in a more general context results that appeared previously in more specialized contexts in work of Klaus Hulek and work of Catherine O’Neil. We give an elementary proof that the commutator pairing on the Heisenberg group corresponds to the Weil pairing on J_{C}[n]. We describe intriguing hyperplane configurations and relate them to the points of hyperosculation on the curve of genus 1.

Contents

Statement by Author 3

Acknowledgments 4

Abstract 5

Chapter I. Introduction 9

I.1. The motivating problem 9

I.1a. Previous work and related work 9 The classical approach 9 The O’Neil approach 10 The Anderson approaches 10

I.1b. Present contribution 10 I.1c. Roadmap for the algorithm 11

I.2. Future research projects 11 I.2a. The family F 11 I.2b. The “j-invariant” approach 11 I.2c. Arithmetic complexity bounds 11 I.2d. Elucidating the hyperplane configurations 12 I.2e. Handling the modular case 12 I.2f. Examples in characteristic p > 0 12

I.3. Chapter-by-chapter overview 12 I.3a. Chapter I: Introduction 12 I.3b. Chapter II: Background on quasi-elliptic curves 12 I.3c. Chapter III: Projective quasi-elliptic curves C [figure: n] P_{k}^{n−1} 13 I.3d. Chapter IV: An algorithm for the jacobian 14 I.3e. Chapter V: Example: a Selmer cubic 15 I.3f. Chapter VI: Example: a pair of quadrics 15 I.3g. Appendix A: Facts about curves of genus 1 15 I.3h. Appendix B: Maps to projective space: a coordinate-free approach 15

I.4. Table of notation 16

Chapter II. Background on quasi-elliptic curves 17 II.1. Assumptions 17 II.2. Definition of a quasi-elliptic curve (C, D) 17

II.2a. Osculating divisors and points of hyperosculation 18 II.2b. Structure of morphisms 19

II.3. Quasi-elliptic curves as maps C [figure: (n)] J_{C} 20

II.3a. Relationship with n-coverings of an elliptic curve 21 II.4. Mapping a quasi-elliptic curve to projective space 21

II.4a. Extending a morphism between quasi-elliptic curves to a morphism between projective spaces 22 Chapter III. Projective quasi-elliptic curves C [figure: n] P_{k}^{n−1} 24

III.1. Attempting to simplify the description 24 III.2. Basic facts about C [figure: n] P_{k}^{n−1} 25 III.3. The saturated homogeneous ideal of C 25 III.4. Projective normality 26 III.5. The Hilbert function of C 28 III.6. C is a complete intersection for n = 3 and n = 4 29 III.7. Cutting C out by quadrics when n ≥ 4 and char(k) ≠ 2 30 III.8. Classifying linear automorphisms of C [figure: n] P_{k}^{n−1} 32 III.9. Concerning the [−1]-automorphisms of C [figure: n] P_{k}^{n−1} 32

III.9a. Procedure for finding the points of hyperosculation 33 III.10. The commutator pairing is the Weil pairing 34

III.10a. Commutator pairings on abelian projective linear groups 34 III.10b. The Weil pairing on an elliptic curve (E, O) 34 III.10c. The commutator pairing on J_{n} is the Weil pairing 35

III.11. The configuration of hyperplanes fixed by J_{n} 37

III.11a. Hyperplane configurations when char(k) ∤ n 39 III.12. Lifting J_{n} ⊂ PGL_{n}(k̄) to H_{n} ⊂ GL_{n}(k̄) 40

III.12a. Case: char(k) ∤ n (i.e., #J_{n} = n^{2}) 40 III.12b. Case: char(k) | n and #J_{n} = n^{2}/p^{e} 41 III.12c. Case: char(k) | n and #J_{n} = n^{2}/p^{2e} 41

III.13. Schrödinger-like representations of H_{n} when char(k) ∤ n 41 III.14. Concerning the Galois module structure of J_{C}[n] 43

Chapter IV. An algorithm for the jacobian 44 IV.1. An algorithm for the jacobian 44 IV.2. Vetting the input 44 IV.3. Describing all matrices that preserve the curve 45 IV.4. Finding H_{n} ⋊ { ±1 } 45

IV.4a. Solving over a field extension 45 IV.4b. Practical improvements for finding H_{n} 46 IV.4c. Practical improvements for finding the [−1] automorphisms 46

IV.5. Finding the points of hyperosculation 46 IV.5a. When n is odd 47 IV.5b. When n is even 47

IV.6. Finding weighted equations for the jacobian 47 IV.7. Obtaining non-weighted equations for J_{C} 49 IV.8. Algorithm summary 49

Chapter V. Example: a Selmer cubic 50 V.1. Vetting the input 50 V.2. Finding H_{3} 50 V.3. Points of hyperosculation and the hyperplane configuration 53 V.4. The curve underlying the jacobian J_{C} 53 V.5. Obtaining a non-weighted model for J_{C} 57 V.6. Musings 57

V.6a. Finding the 2-torsion on (C, O) 57 V.6b. Obtaining a Weierstrass model for J_{C} 58 V.6c. The classical Hessian 58

V.7. Tackling the family ax^{3} + by^{3} + cz^{3} + mxyz = 0 59 V.7a. History 59 V.7b. Finding the jacobian by the algorithm 59 V.7c. Applying the Riemann–Roch algorithm 60 V.7d. Comparing with result from classical invariant theory 61

Chapter VI. Example: a pair of quadrics 62 VI.1. The curve 62 VI.2. Finding the [−1] matrices 65 VI.3. Finding a point of hyperosculation 65 VI.4. The invariant theory of H_{4} 66

Appendix A. Facts about curves of genus 1 67

A.1. Consequences of Riemann–Roch on curves of genus 1 67 A.1a. The canonical divisor and differential forms 67 A.1b. Dimension of complete linear systems 67 A.1c. No two points are linearly equivalent 68 A.1d. Effective representatives 68

A.2. The group law 68

A.2a. Isogenies 69 A.3. The jacobian action 69

A.3a. Torsion packets on curves of genus 1 69 A.4. Morphisms 70

A.4a. Morphisms between curves of genus 1 70 A.4b. Endomorphisms of curves of genus 1 71 A.4c. Automorphisms of curves of genus 1 71 A.4d. Separable and inseparable degree of a morphism 73 A.4e. The degree and fixpoints of a curve endomorphism 73 A.4f. Correspondence between morphisms and subgroups of J_{C} 74

A.5. Consequences of Riemann–Hurwitz on curves of genus 1 75

Appendix B. Maps to projective space: a coordinate-free approach 76

B.1. Background material 76

B.1a. Projective space bundles 76 Points correspond to rank 1 quotients 77 Change of base 77 Coordinates 77

B.1b. Starting with a free A-module 77 B.1c. When the base is a field 78

B.2. Coordinate-free version of X [figure] P_{X}^{n} 78 B.3. Mapping properties 79

B.3a. Morphism of schemes 80 B.3b. Morphism of sheaves 80 B.3c. Different subspaces of H^{0}(X, L) 80

B.4. Very ample invertible sheaves 81

Bibliography 82

CHAPTER I

Introduction

I.1. The motivating problem

The motivation behind this work is the following arithmo-geometric problem: given equations defining a curve C of genus 1 over a perfect field k, find equations defining the jacobian curve J_{C}. The arithmetic nature of the problem is reflected in its triviality when k is algebraically closed, for in that case we have C ≅ J_{C}. (The problem also turns out to be trivial when k is a finite field—see §I.2f.)

This problem is in some sense the opposite or inverse of the following problem, which occurs in the theory of descent: given an elliptic curve, produce its principal homogeneous spaces.

I.1a. Previous work and related work

Let n be the degree of a k-rational divisor on C. The n = 1 case of the motivating problem is trivial: we have C ≅ J_{C}.

For n ≥ 2, it turns out the curve C always admits a non-degenerate degree n map to P_{k}^{n−1}, so without loss of generality we may assume that C is given that way to begin with. For n ≥ 3 that map is an embedding, so we are essentially assuming that C is given to us as a non-degenerate degree n curve in P_{k}^{n−1} of genus 1.

The classical approach. The cases n = 2, n = 3, n = 4, and n = 5 of the motivating problem, or slight variants thereof, have been solved, under the hypothesis char(k) ∤ n (in some cases also char(k) ≠ 2, 3), using techniques of classical invariant theory: [Wei54], [MSS96], [AKM^{+}01], [VT], and [Fis]. The first four references rely on results from invariant theory worked out in the 19th century, and thus amount to establishing interesting modern interpretations of classical results. The work [Fis], on the other hand, does quite a bit more: it develops from scratch the relevant invariant theory—which does not appear classically—and then gives the interpretation in terms of jacobians.

This “classical approach” requires one to work with a suitable family of curves of genus 1. The classical invariant theory of the generic equations defining the family describes a fundamental system of covariants and their relations, called syzygies. In the cases considered so far, there has always been one Weierstrass-like syzygy (I am not aware of a satisfactory explanation for that), which turns out to define the jacobian J_{C} of each member C of the family.

However, we know of a family F (see §I.2a), for which the classical approach appears to go through without a hitch, yet the Weierstrass-like syzygy turns out to define a quadratic twist of the jacobian, not the jacobian itself! A little extra work then identifies which twist occurred, thereby leading to the actual jacobian.

Because the classical approach works with a family of curves, it ends up producing formulas for the coefficients of J_{C} in terms of the coefficients of C. The approach itself does not apply to individual curves, but once the formulas are known, they can of course be applied to individual curves, giving us an algorithm (namely: substitution) of trivial computational complexity for solving a restricted version of the motivating problem; namely, its restriction to the particular curves that occur in the family under consideration.

The O’Neil approach. With additional assumptions on the starting curves and the ground field k, the motivating problem is treated by an intriguing approach in [O’N01]: the jacobian J_{C}, together with a level-n structure (this is part of the extra data on C), is expressed as a point on the modular curve X_{1}(n). For n = 3 and n = 5, [O’N01] exhibits formulas for J_{C} ∈ X_{1}(n) in terms of C.

The Anderson approaches. A related problem has been studied by Greg Anderson: given a curve of genus g defined over an algebraically closed field, find equations defining its jacobian (an abelian variety of dimension g). One approach to this problem, which may lead to an algorithm, is given in [And02]. A different, and algorithmic, approach is given in [And]. By assuming the ground field to contain symbols that are algebraically independent over the prime subfield, this approach can treat a family of curves of genus 1, which then leads to formulas for the jacobian (which can then be applied to individual curves). Even though the formulas are defined over the prime subfield, it is unclear whether they always give the jacobian over that field—certainly they do so over the field’s algebraic closure. As shown in [And], when the Anderson algorithm is applied over Q to two of the cases considered in [AKM^{+}01], it produces the same equations for the jacobian as exhibited in [AKM^{+}01], and thus gives the correct answer over Q.

I.1b. Present contribution

In the present work, again assuming char(k) ∤ n (cf. §I.2e), we develop an algorithm (a computational process guaranteed to complete in a finite number of computational steps), based on ideas given to us by Minhyong Kim, that solves the motivating problem in all degrees n ≥ 3.

Unlike the approaches mentioned previously, this algorithm can be applied directly to individual curves. Of course, it can then also be applied to a family, serving as an alternate approach for obtaining formulas for J_{C}. (For example, in chapter V we apply the algorithm to the individual curve 3x^{3}+4y^{3}+5z^{3} = 0, and then show in §V.7 how to tackle the family ax^{3}+by^{3}+cz^{3}+mxyz = 0.)

After applying our algorithm, if one desires a Weierstrass model for J_{C}, one could apply the usual Riemann–Roch algorithm (cf. §V.7c): find functions x, y with poles of orders 2 and 3 at the origin of J_{C}, then use linear algebra to find the relation between 1, x, y, x^{2}, xy, y^{2}, x^{3}.

The mere existence of an algorithm is not surprising, at least when k is countable, for there certainly exists the following type of exhaustive search: given C, one enumerates all varieties defined over k (there are countably many); for each such variety V, one enumerates all k-rational points in the ambient space; for each such pair (V, P), one enumerates all algebraic maps from the ambient space of C to the ambient space of V; for each such triple (V, P, ϕ), one checks whether V is a curve of genus 1, whether P lies on V, and whether ϕ has degree n^{2} and the difference of any pair of points in ϕ^{−1}(P) is a divisor class of order dividing n in the divisor class group; all of this can be done algorithmically, and once all the conditions are met, by [AKM^{+}01, Prop. 4.5], one has found J_{C}. (This also shows that the motivating problem lies in the computational complexity class np: a nondeterministic polynomial-time algorithm is to guess and then verify the certificate data indicated above.) But exhaustive search is completely useless: even in the simplest of cases, it exhausts our patience, and it has the philosophical drawback that its description relies only peripherally on the mathematics of the motivating problem.

In contrast to exhaustive search, the algorithm in the present work relies heavily on the mathematics of the motivating problem, and the algorithm can be successfully applied in practice (as we demonstrate in chapters V and VI). However, on multiple occasions in the algorithm, we need to find all solutions to a 0-dimensional intermediate problem; unfortunately, today we know how to complete those steps algorithmically only by appealing to the theory of Gröbner bases, and thus the overall algorithm, in its present form, has seemingly unbounded computational complexity.

In summary, our present algorithm has the advantage over the previous approaches of being applicable both to families and to individual curves, and of course the advantage that it can be applied in all degrees n ≥ 3. On the other hand, from a practical perspective, on a current desktopclass computer (3 GHz Pentium processor, 2 GB RAM), we have been successful only with particular instances in the n = 3 and n = 4 cases of the motivating problem.

I.1c. Roadmap for the algorithm

As described below in the chapter-by-chapter overview (see §I.3), this dissertation contains more than just the algorithm that solves the motivating problem. If the reader’s only goal is to understand that algorithm, then the summary below in §I.3d may be a good starting point, and the more detailed roadmap is as follows.

In chapter II, the only crucial material is: §II.2 on the map j_{D} and the isomorphism C/J_{C}[n] [figure: ∼] J_{C}, and §II.2a on the points of hyperosculation. In chapter III, the only crucial material is: §III.8 on classifying linear automorphisms, §III.11 on the fixpoints and hyperplane configurations associated to generators of J_{n}, and §III.12 on lifting J_{n} ⊂ PGL_{n}(k̄) to H_{n} ⊂ SL_{n}(k̄). Of course, all of chapter IV (description of the algorithm) is crucial, and the examples from chapter V and VI serve as concrete illustrations of the steps.

I.2. Future research projects

I.2a. The family F

Consider the family F: Z^{3} = U(X, Y), where U is a generic binary cubic form with distinct roots. When we look at the classical invariant theory of binary cubic forms (cf. [Stu93, 3.7.6, 3.7.7]), we find a Weierstrass-like syzygy. Let E be the elliptic curve over Q defined by that syzygy, and let C be a member of the family F with coefficients in Q. The syzygy leads to a map C [figure]E of degree 3. Thus E cannot be J_{C}, for a map C [figure] J_{C} defined over Q must have degree a square: its pullback is multiplication-by-m for some m. It turns out E is the quadratic twist of J_{C} over Q(\sqrt{−3}), which is the field over which J_{C} has complex multiplication. Further investigations into F may give additional insight into the classical approach: when it works, and why it works when it does. (I intend to continue this research and publish the results in the future.)

I.2b. The “j-invariant” approach

An entirely different approach to finding J_{C}, not discussed elsewhere in this dissertation, goes as follows: starting with C defined over k, move to a finite field extension K/k over which C_{K} admits a point, then apply the Riemann–Roch algorithm (cf. §V.7c) to obtain a Weierstrass model W for C_{K} that has k-rational coefficients. (From this we can read off the j-invariant of C—my original motivation for this approach—whence the name.) Thus C, W, and J_{C} (as an abstract curve) are all defined over k, and all three are isomorphic over K. But W and J_{C} are elliptic curves over k, whence either W ≅ J_{C}, or W is a quadratic twist of J_{C} over one of the quadratic extensions lying inside K/k. By writing down the possibilities, we obtain a finite list of Weierstrass models that are candidates for J_{C} (and one of them must actually be J_{C}). Next repeat the procedure with a different field extension K/k, hopefully obtaining a different finite list of candidates that necessarily contains J_{C}. Repeat this process until the various finite lists intersect in a single candidate, which must then be J_{C}. (I have successfully applied this approach in the case n = 2. I intend to continue this research and publish the results in the future.)

Compared to the classical approach, or to the Anderson algorithm, or to the algorithm in this dissertation, the “j-invariant” approach has the following disadvantage: it does not give you a map C [figure] J_{C}.

I.2c. Arithmetic complexity bounds

Any systematic approach to the jacobian might lead to relations between the arithmetic height (or other suitable notion of complexity) of J_{C} and the height of C. (A preliminary investigation of the “j-invariant” approach led to rough height relations. I intend to continue this line of research in the future.)

I.2d. Elucidating the hyperplane configurations

We feel that more can be said about fixpoints and fixed hyperplanes of elements of PGL_{n}(k̄) that preserve C (cf. §III.9 and §III.11). For example, what can be said about the hyperplanes fixed by the [−1]-automorphisms? For another example, we discover in chapter V that the [−1]-automorphisms have eigenvalue +1 at one fixpoint but eigenvalue −1 at the other three fixpoints. Why does this happen, and what happens in general? Many such questions are easily formulated from the examples worked out in chapters V and VI.

It would also be nice to have a formula, or at least an algorithm more elegant than brute force, for determining the osculating hyperplane at a point (thus also for determining the hyperosculating hyperplanes at the points of hyperosculation).

A better understanding of these matters may lead to further simplifications of the elementary proof in §III.10 that the commutator pairing is the Weil pairing, or at least some kind of insight into what is happening geometrically in each step of the proof.

I.2e. Handling the modular case

Our algorithm for the jacobian exploits the isomorphism (∗) (from §I.3b below), which holds in general. Even the recasting of C/J_{C}[n] as C/H_{n} (cf. §I.3d below) holds in general. However, to compute C/H_{n} in practice, we start by finding the k̄-valued points of the group scheme H_{n}. When char(k) | n (known as the modular case), we have #H_{n} ≠ #H_{n}(k̄), and our algorithm breaks down. (Note that the invariant theory of finite groups itself is not an obstacle: algorithms for the modular case appear in [Kem96] and [DHS98].) However, since we do determine equations defining the group scheme H_{n}, even in the modular case there may be a method for determining equations for C/H_{n}.

I.2f. Examples in characteristic p > 0

It would be interesting to apply our algorithm to examples in characteristic p > 0. For example, whereas working over function fields in characteristic 0 may be computationally out of reach on today’s computers, it may turn out that something similar is feasible in characteristic p.

It is worth remarking that the motivating problem is actually trivial when k is the finite field F_{q}, where q is a power of p. By the Riemann hypothesis for curves (cf. [Har77, Ex. V.1.10]), we can easily see that a curve of genus g defined over F_{q} necessarily admits a rational point when q exceeds a bound expressible in terms of g. For g = 1, the condition is q ≥ 2. Therefore, our curve C of genus 1 always admits an F_{q}-rational point, whence C ≅ J_{C}. But this isomorphism is not canonical; in particular, given a family of curves of genus 1, it is unclear whether we can pick out the rational point in a consistent manner across the family. In other words, a non-canonical isomorphism C ≅ J_{C} exists for each individual members of the family, but there may be no single such isomorphism for the entire family. Our algorithm, on the other hand, always produces J_{C} in a canonical fashion. Thus, it is still interesting to apply the algorithm even when working over a finite field.

I.3. Chapter-by-chapter overview

I.3a. Chapter I: Introduction

You’re reading that chapter right now!

I.3b. Chapter II: Background on quasi-elliptic curves

Interest in the motivating problem led us to study an object we call a quasi-elliptic curve defined over k: a pair (C, D), where C is a curve of genus 1 defined over k, and D is a k-rational divisor class on C. Each quasi-elliptic curve admits precisely n^{2} (when char(k) ∤ n) points of hyperosculation, which are k̄-valued points P such that nP ∈ D. They compose a k-rational orbit under the action of the n-torsion J_{C}[n] on C. This action respects the fibers of the map j_{D}: C [figure] J_{C}, P [figure] [nP] − D, and we get an isomorphism of curves

\frac{C}{J_{C}[n]} [figure: ∼] J_{C}.  (∗)

The hyperosculation orbit on C is the k-rational point on the left that goes to the origin of J_{C} on the right.

Remark. Our algorithm for J_{C} will in fact find equations for C/J_{C}[n]; it will find coordinates for the points of hyperosculation on C, and thus obtain the coordinates of the origin on C/J_{C}[n].

Chapter II proceeds with a description of morphisms between quasi-elliptic curves, and in particular establishes

Aut(Ck̄, D) = J_{C}[n](k̄) ⋊ Aut(Ck̄, O),  (∗∗)

where O ∈ C(k̄) is a point of hyperosculation.

We next give an alternative description of a quasi-elliptic curve, namely as a map C [figure]J_{C}, such as j_{D} above, with the property that the induced pullback map J_{C} [figure] J_{C} is multiplication-by-n. While working on the paper [AKM^{+}01], such maps occurred when we were considering new ways of describing principal homogeneous spaces and n-coverings of an elliptic curve. We briefly give those descriptions here.

Finally, when the class D admits a k-rational representative D, so that we are working with a pair (C, D), there is always the associated non-degenerate degree n map to the projective space bundle P(H^{0}(C, O(D))), which upon choosing coordinates is just P_{k}^{n−1}. (The map is an embedding for n ≥ 3.) We show that automorphisms of (C, D) extend to automorphisms of P_{k}^{n−1}.

I.3c. Chapter III: Projective quasi-elliptic curves C [figure: n] P_{k}^{n−1}

This chapter generalizes material on “elliptic normal curves” from [Hul86] and on “n-prepared curves” from [O’N01].

We study pairs (C, D) with n ≥ 3 (cf. the last paragraph in the previous section), but identify them with their image in P_{k}^{n−1}; thus, we study “non-degenerate degree n curves in P_{k}^{n−1} of genus 1” (it turns out, as we show, that none of these words is superfluous), and denote such a curve C [figure: n] P_{k}^{n−1}.

We show that C [figure: n] P_{k}^{n−1} is projectively normal, we compute its Hilbert function and Hilbert polynomial, and we show that each generating set for the saturated ideal defining the curve contains n(n − 3)/2 linearly independent quadratic forms. Furthermore, when char(k) ≠ 2, for n ≥ 4 each minimal generating set comprises precisely n(n − 3)/2 quadratic forms; that is, C is cut out schemetheoretically by quadrics.

Combining (∗∗) with the last paragraph in §I.3b, we obtain

{ ϕ ∈ PGL_{n}(k̄) : ϕ(Ck̄) = Ck̄ } ≅ J_{C}[n](k̄) ⋊ Aut((Ck̄, O)),  (∗∗∗)

and from this we conclude: the subgroup J_{n} ⊂ PGL_{n}(k̄) of elements that preserve C and act fixpoint-free on C is a faithful and Galois equivariant representation of the action of J_{C}[n](k̄) on C.

No matter the characteristic of k or the j-invariant of C, the part of (∗∗∗) that always exists is J_{n} ⋊ {±1}. The non-trivial coset of J_{n} is characterized as the subset of PGL_{n}(k̄) whose elements preserve C, have a fixpoint on C, and have order 2. When n is odd, each point of hyperosculation on C occurs as the fixpoint of an element of that coset. When n is even, either all or none of the fixpoints of an element of that coset are points of hyperosculation. We can identify whether a given fixpoint is a point of hyperosculation by searching for a hyperplane that meets C with multiplicity n at the given point. Thus we have a practical procedure for finding the points of hyperosculation on C.

By choosing lifts from PGL_{n}(k̄) to GL_{n}(k̄), we define the commutator pairing

\begin{matrix}J_{n} × J_{n} & [figure] k̄^{×}, \\ ([M], [N]) & [figure] M N M^{−1}N^{−1}. \end{matrix}

Using the points of hyperosculation as a crutch, we give an elementary proof that the commutator pairing on J_{n} corresponds to the Weil pairing on J_{C}[n](k̄). To even discuss the Weil pairing, we must assume char(k) ∤ n, and this assumption now occurs repeatedly as we appeal to the commutator pairing and the fact that J_{n} has two generators: we show that lifts of generators M and N for J_{n} have distinct eigenvalues, that each generator cyclically permutes the fixpoints (eigenspaces) of the other, and the same is true of the fixed hyperplanes (eigenspaces of the transposed matrices).

For n odd, this leads to the following hyperplane configuration: the hyperplanes fixed by M intersect C in the n^{2} points of hyperosculation, and the same is true of the hyperplanes fixed by N. Furthermore, each point of hyperosculation lies in the intersection of a unique hyperplane fixed by M with a unique hyperplane fixed by N. (This gives yet another procedure for finding the points of hyperosculation on C.)

For n even, the hyperplanes fixed by M intersect C in a collection of n^{2} points that, under the isomorphism (∗), correspond to a non-trivial point of 2-torsion on J_{C}, while the hyperplanes fixed by N do the same thing, but for a different point of 2-torsion on J_{C}.

Next, we define H_{n} ⊂ SL_{n}(k̄) to be the preimage of J_{n} under the map SL_{n} [figure] PGL_{n}, giving us the exact sequence

1 [figure] μ_{n}(k̄) [figure] H_{n} [figure] J_{n} [figure] 1.

This is a central extension of J_{n} by μ_{n}(k̄), of order n^{3} (when char(k) ∤ n). We characterize the different lifts of J_{n} to GL_{n}(k̄) of order n^{3}, and establish that H_{n} is essentially the only lift we would want to work with.

We finish the chapter by describing Schrödinger-like representations for H_{n} and how they relate to the Galois module structure of J_{C}[n].

I.3d. Chapter IV: An algorithm for the jacobian

We assume C to be smooth and given as in §I.3c: a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. We furthermore assume char(k) ∤ n (cf. §I.2e). Actually, the algorithm takes arbitrary equations as input, and then verifies that the input meets the assumptions. Thus

C = Proj \frac{k[x_{0}, …, x_{n−1}]}{I},

where the generators of I were given in the input.

Now we exploit (∗). As explained in §I.3c, the action of J_{C}[n](k̄) on C(k̄) is given by the group J_{n} ⊂ PGL_{n}(k̄), which we lift to H_{n} ⊂ SL_{n}(k̄). We find H_{n} as follows: using a Gröbner basis for the ideal of C, we easily write down the conditions for a generic matrix to preserve C and have determinant 1. In finding the finitely many solutions, we obtain a finite field extension K/k over which each solution is defined. We can then check which matrices have no fixpoints on C, and which have fixpoints and are of order 2 in PGL_{n}(k̄). Thus we can find matrices representing the elements of J_{n} ⋊ {±1}. We identify two generators M, N for H_{n} by searching for a pair of matrices whose commutator is a primitive nth root of unity.

By finding eigenvectors for each matrix and its transpose, we obtain the fixpoints and fixed hyperplanes under the action of the induced element of PGL_{n}(k̄). As explained in §I.3c, this information gives us the coordinates of the points of hyperosculation.

We identify H_{n} with the K-valued points of a finite group scheme, again denoted H_{n}, which is defined over k. Then (∗) becomes

J_{C} ≅ \frac{C}{H_{n}} = Proj ((\frac{K[x_{0}, …, x_{n−1}]}{K I})^{H_{n}(K)})^{Gal(K/k)}

= Proj \frac{k[x_{0}, …, x_{n−1}] ∩ K[x_{0}, …, x_{n−1}]^{H_{n}(K)}}{I ∩ K[x_{0}, …, x_{n−1}]^{H_{n}(K)}} .

Algorithms in the invariant theory of finite groups (cf. [Stu93]) tell us how to find the invariants under H_{n}(K). Because H_{n} itself is invariant under Gal(K/k), these algorithms already give us the correct answer over k.

Finally, substituting the coordinates of one of the points of hyperosculation on C gives us the coordinates of the k-rational origin on J_{C}.

I.3e. Chapter V: Example: a Selmer cubic

To illustrate our theory in the case n = 3, we apply the algorithm from chapter IV to the Selmer cubic

C = Proj \frac{Q[x, y, z]}{⟨3x^{3} + 4x^{3} + 5z^{3}⟩}.

We show how to carry out each step of the algorithm by hand (but also include computer code in some cases), showing in moderate detail how to apply the well-known intermediate algorithms that were referenced in chapter IV.

At the end of the chapter, we rely on the earlier results in this chapter to tackle the family

Proj \frac{k[x, y, z]}{⟨ax^{3} + bx^{3} + cz^{3} + mxyz⟩},

where k is a perfect field with char(k) ≠ 3.

We include example code for the computer systems we used: GP/Pari [BBB^{+}00], Macaulay 2 [GS], Mathematica [Wol03], mwrank [Cre], and Singular [GPS01].

I.3f. Chapter VI: Example: a pair of quadrics

To illustrate the case n = 4, we carry out some of the steps of applying the algorithm from chapter IV to

C = Proj \frac{Q[w, x, y, z]}{⟨w^{2} + x^{2} + y^{2} + z^{2}, w^{2} + 2x^{2} + 3y^{2} + 4z^{2}⟩}.

We include computer code that is easily adapted to arbitrary values of n.

Compared to the example from the previous chapter, the current example is interesting for at least two reasons: it illustrates how the algorithm works when n is even, and the finite Heisenberg group H_{4} looks even less Schrödinger-like.

I.3g. Appendix A: Facts about curves of genus 1

We state facts about curves of genus 1 that we reference in other parts of this dissertation.

I.3h. Appendix B: Maps to projective space: a coordinate-free approach

Well known from the theory of smooth projective curves is that a divisor D on a curve X leads to the map X [figure] P^{n} given by P [figure] [s_{0}(P) : …: s_{n}(P)], where the s_{i} are elements of L(D) = H^{0}(X, O(D)).

In this appendix, for lack of a suitable reference, we give a coordinate-free description of such maps. (For the same material described in terms of coordinates, see [Har77, §II.7].)

We describe projective space bundles, how to think of their points as certain types of rank 1 quotients, and how they behave under base change. For X a noetherian scheme over a noetherian ring A, and L an invertible sheaf on X that is generated by global sections and such that H^{0}(X, L) is a free A-module of finite rank, we obtain a canonical morphism

ϕ: X [figure] P(H^{0}(X, L) ⊗_{A} O_{A}),

from X to the displayed projective space bundle. When A is a field k, and K is a field extension of k, the map ϕ has a nice description in terms of K-valued points:

\begin{matrix}ϕ_{V} : X(K) & [figure] P(H^{0}(X, L))(K), \\ P & [figure] \{ s ∈ H^{0}(X, L) ⊗_{k} K : s_{P} = 0 \}. \end{matrix}

In other words, ϕ carries the K-valued point P to the hyperplane, in the vector space H^{0}(X, L)⊗_{k} K, of global sections that vanish at P.

16 I. Introduction

I.4. Table of notation

Symbol  Explanation
k, k̄, G_{k}  perfect field, algebraic closure, absolute Galois group Gal(k̄/k)
K  field extension of k, often a finite and/or Galois extension in k̄
P_{k}^{n−1}  projective (n − 1)-space over k ≡ Proj(k[X_{0}, …, X_{n−1}])
P(·)  projective space bundle ≡ space of hyperplanes
X  a k-scheme, often a k-curve of arbitrary genus
X_{K}  the K -scheme X ×_{k} K
X(K )  the K -valued points of X ≡ morphisms Spec(K ) [figure] X
C  a k-curve of genus 1
k(C)  field of k-rational rational functions on C(k̄)
J_{C}  the jacobian of C
D  k-rational divisor class on C of positive degree
n  degree of D
p  char(k)
e  when char(k) | n, we have p^{e} | n but p^{e+1} ∤ n
j_{D}  the map C [figure] J_{C}, P [figure] [nP] − D
H  hyperosculation packet on C (see II.2a)
D  k-rational divisor on C of positive degree
[D]  set of all divisors linearly equivalent to D
[D]_{k}  set of k-rational divisors linearly equivalent to D
(C, O)  elliptic curve with group law origin O ∈ C(k)
(C, D)  quasi-elliptic curve (see II.2)
C [figure: (n)] J_{C} map whose pullback J_{C}  [figure] J_{C} is multiplication-by-n (see II.3)
C [figure: n] P_{k}^{n−1} non-degenerate degree n curve in P_{k}^{n−1} of genus 1 (see III.1)
I  saturated homogeneous ideal of C (see III.3)
J_{n}  J_{C}[n](k̄) disguised as a certain subgroup of PGL_{n}(k̄) (see III.10)
H_{n}  finite Heisenberg group in SL_{n}(k̄) covering J_{n} (see III.12)
𝔊  Gröbner basis for an ideal

Table I.1. Notation used throughout this dissertation.

CHAPTER II

Background on quasi-elliptic curves

II.1. Assumptions

Let k be a perfect field, and throughout let k̄ denote an algebraic closure. Without further context or qualification, use of “Galois” refers to the absolute Galois group G_{k} := Gal(k̄/k).

A k-variety is a separated, geometrically integral, finite-type k-scheme, and a curve is a variety of dimension 1. If X is a k-scheme, and K ⊇ k is a field extension, then X_{K} denotes the K -scheme

X_{K} := X ×_{k} K := X ×_{Spec(k)} Spec(K ).

Thus a k-variety may be defined as a finite-type k-scheme X such that the base change Xk̄ is a variety in the sense of classical algebraic geometry carried out over algebraically closed fields.

Throughout, C is a k-curve of genus 1, and J_{C} denotes the jacobian of C. Basic facts about curves of genus 1 appear in appendix A. Crucial among them is the fact, which follows from Riemann–Roch, that each K -rational divisor class of degree 1 admits a unique K -rational point as representative (for details, see §A.1d); from this, we can give an elementary definition both of the canonical group law on an elliptic curve (see §A.2) and of the canonical action of J_{C} on C (see §A.3). Familiarity with these facts is assumed.

II.2. Definition of a quasi-elliptic curve (C, D)

Recall that an elliptic k-curve is a pair

(C, O),

where C is a k-curve of genus 1 and O is a k-rational point on C. A morphism of elliptic curves (C, O) [figure](C′, O′) is a morphism of curves ϕ: C [figure]C′ so that ϕ(O) = O′. In fact, (C, O) can be given a canonical group scheme structure; in particular, for any field extension K ⊇ k, the set C(K ) is a group with identity element O, and any morphism of elliptic curves is necessarily a group homomorphism on K -valued points. Associated with (C, O) is the canonical k-morphism j_{O}: C [figure] J_{C} of degree 1, determined on k̄-valued points by

\begin{matrix}j_{O}: C(k̄) & [figure: ∼] J_{C}(k̄), \\ P & [figure] [P − O]. \end{matrix}

Via this map, which evidently takes the group law origin on C to the group law origin on J_{C}, we have: every elliptic curve is canonically isomorphic to its jacobian.

Note that a given k-curve C of genus 1 need not occur as an elliptic k-curve, since C need not admit a k-valued point.

In analogy with the above, we define a quasi-elliptic k-curve to be a pair

(C, D),

where C is a k-curve of genus 1 and D is a k-rational divisor class on C of positive degree. (In §II.4, and throughout chapter III, we’ll further assume that D admits a k-rational representative, but for now it is only the class that is k-rational.) A morphism of quasi-elliptic curves (C, D) [figure] (C′, D′) is a morphism of curves ϕ: C [figure] C′ so that ϕ_{∗}D = D′. We define the degree of (C, D) to be the degree of D. Let n be that degree. Associated with (C, D) is the canonical k-morphism j_{D}: C [figure] J_{C} of degree n^{2}, determined on k̄-valued points by

\begin{matrix}j_{D}: C(k̄) & [figure] J_{C}(k̄), \\ P & [figure] [nP] − D. \end{matrix}

Note that each k-curve C of genus 1 does occur as a quasi-elliptic k-curve: simply define a k-rational divisor class on C by taking the Galois orbit of any k̄-valued point of C.

On a curve of genus 1, since each k-rational divisor class of degree 1 has a unique k-rational point as representative, a quasi-elliptic curve of degree 1 is the same thing as an elliptic curve.

II.2a. Osculating divisors and points of hyperosculation

On a curve C of genus 1, the canonical action of J_{C} on C induces, for each integer n ≥ 1, an action of J_{C}[n](k̄) on C(k̄). The orbits under this action are called n-torsion packets. Here are the basic facts (explained in more detail in §A.3a):

• An n-torsion packet is a maximal collection of points in C(k̄) so that, if P, Q are any two of them, then there is a linear equivalence nP ∼ nQ.

• The set of n-torsion packets on C is in one-to-one correspondence with the set (C/J_{C}[n])(k̄).

• Given an n-torsion packet P ⊂ C(k̄), we can pick O ∈ T and consider the elliptic curve E = (Ck̄, O). Then P comprises the points of n-torsion: P = E[n](k̄). Thus an n-torsion packet is a collection of points which would be the (usual) n-torsion were we to choose one of them as group law origin.

• The Galois conjugate of an n-torsion packet is again an n-torsion packet.

Definition. Let (C, D) be a quasi-elliptic curve of degree n. A point P ∈ C(k̄) such that nP ∈ D is called a point of hyperosculation. For each P ∈ C(k̄), the unique divisor of the form (n−1)P +Q that lies in D is called the osculating divisor at P.

The uniqueness is explained as follows: since the divisor class D − (n − 1)[P] has degree 1, it is represented by a unique point Q ∈ C(k̄). Observe that a point of hyperosculation may be characterized as one whose osculating divisor involves that point more than usual.

Remark. The terms “osculation” and “hyperosculation” have a geometric interpretation for n ≥ 3. Identify C with its image in P_{k}^{n−1} via the embedding given by D (see §II.4). Associated with each P ∈ C(k̄) is the hyperplane meeting C to maximal order at P, called the osculating hyperplane. For most P, we have just seen that the osculating hyperplane meets C to order n − 1 at P. For certain points P, the osculating hyperplane meets C to order n at P, and that phenomenon is called hyperosculation. A point whose osculating plane hyperosculates is called a point of hyperosculation. For a cubic curve in P_{k}^{2}, the osculating hyperplane at each point is simply the tangent line. The 9 points of hyperosculation are simply the flex points—they are the points where the tangent line meets the curve to order 3.

Proposition II.2.1. On a quasi-elliptic curve (C, D), the points of hyperosculation compose an ntorsion packet. There are precisely #J_{C}[n](k̄) distinct points of hyperosculation in C(k̄). (Therefore, when char(k) ∤ n, there are n^{2} such points.)

Proof. It is easy to check that the points of hyperosculation are one orbit under the action of J_{C}[n](k̄) on C(k̄). □

Corollary II.2.2. Every quasi-elliptic curve has at least one point of hyperosculation. (It might not be k-rational, but see II.2.3.)

Definition. The n-torsion packet H comprising points of hyperosculation is called the hyperosculation packet.

Proposition II.2.3. The hyperosculation packet H is Galois stable.

II.2. Definition of a quasi-elliptic curve (C, D) 19

Proof. If P is such that nP ∈ D, then it is easy to see that P^{σ} has the same property: nP^{σ} = (nP)^{σ} ∈ D^{σ} = D. □

The space of n-torsion packets is (C/J_{C}[n])(k̄). If C is an arbitrary k-curve of genus 1, then C/J_{C}[n] need not admit a k-rational point; but, by II.2.3, when C is part of a quasi-elliptic curve (C, D) of degree n, then C/J_{C}[n] always admits a canonical k-rational point, namely the hyperosculation packet H. In other words, (C/J_{C}[n], H) is an elliptic curve.

Theorem II.2.4. The k-morphism j_{D} of degree n^{2}, given on k̄-valued points by

\begin{matrix}j_{D}: C(k̄) & [figure] J_{C}(k̄) \\ P & [figure] [nP] − D, \end{matrix}

descends to a k-isomorphism of elliptic curves

j_{D}: C/J_{C}[n] [figure: ∼] J_{C}.

(When n = 1, this is just the usual isomorphism C [figure: ∼] J_{C}.)

Proof. To see this, observe merely that the map has degree 1, and that the fibers of j_{D} are the same as the orbits under the action of J_{C}[n]; in particular, the hyperosculation packet is j_{D}^{−1}(0). □

II.2b. Structure of morphisms

The definition of a morphism between two quasi-elliptic curves was given in §II.2. For there to exist such a morphism, it is of course necessary that the two quasi-elliptic curves have the same degree. We will now give descriptions of the set of morphisms between two quasi-elliptic curves, of the monoid of endomorphisms of a quasi-elliptic curve, and of the group of automorphisms of a quasielliptic curve. These particular descriptions require the choice of a point of hyperosculation. Since there may be no such choice that is k-rational, these descriptions may not be useful for describing k-rational homomorphisms, k-rational endomorphisms, and k-rational automorphisms.

Proposition II.2.5. Let (C, D) and (C′, D′) be two quasi-elliptic curves of the same degree n. Fix points of hyperosculation O ∈ C(k̄) and O′ ∈ C′(k̄). Each morphism

ψ ∈ Hom((Ck̄, D), (C′ ; k̄, D′))

can be written uniquely in the form

ψ = τ ∘ ϕ, where τ ∈ J_{C′}[n](k̄), ϕ ∈ Hom((Ck̄, O), (C′ ; k̄, O′)).

Moreover, vice versa, each such τ ∘ ϕ defines a ψ.

Proof. By A.4.1, we certainly can write ψ = τ ∘ ϕ with ϕ as above and τ ∈ J_{C′}(k̄). Then ψ_{∗}(nO) = τ_{∗}ϕ_{∗}(nO) = τ_{∗}(nO′) = nτ(O′), and we have nτ(O′) ∼ nO′ if and only if n(τ(O′) − O′) ∼ 0, i.e., if and only if τ has order n. □

Corollary II.2.6. Let (C, D) be a quasi-elliptic curve of degree n. Fix a point of hyperosculation O ∈ C(k̄). Each endomorphism

ψ ∈ End((Ck̄, D))

can be written uniquely in the form

ψ = τ ∘ ϕ, τ ∈ J_{C}[n](k̄), ϕ ∈ End((Ck̄, O)).

Proposition II.2.7. Let (C, D) be a quasi-elliptic curve of degree n. Fix a point of hyperosculation O ∈ C(k̄). The automorphism group of (Ck̄, D) has the structure

Aut((Ck̄, D)) = J_{C}[n](k̄) ⋊ Aut((Ck̄, O)).

Proof. As in II.2.5, this time by A.4.5. □

II.3. Quasi-elliptic curves as maps C [figure: (n)] J_{C}

Associated with the quasi-elliptic curve (C, D) is the map j_{D}: P [figure][nP] − D, which has the property:

the induced map (j_{D})_{∗}: J_{C} [figure] J_{C} is multiplication-by-n.

We will now classify maps with that property and thereby obtain another description of quasi-elliptic curves.

To get started, forget D for the time being, and let ϕ: C [figure] J_{C} be any morphism whatsoever. For each integer n ≥ 1, we associate with ϕ the map

ϕ_{n}: C [figure] Pic_{C}^{n}

whose behavior on k̄-valued points is

ϕ_{n}: P [figure] unique class D_{P} so that ϕ(P) = [nP] − D_{P} .

Proposition II.3.1. We have ϕ_{∗} = [n] if and only if ϕ_{n} is constant; when that is the case, ϕ is defined over k if and only if the value of the constant map ϕ_{n} in Pic_{C}^{n} is a k-rational class.

Proof. If ϕ_{n} is constant, say with value D, then ϕ(P) = [nP] − D for all P ∈ C(k̄), whence ϕ_{∗} is immediately seen to be [n]. Now assume ϕ_{∗} = [n]. Let P, Q ∈ C(k̄) be arbitrary. Then [nP − nQ] = n[P − Q] = ϕ_{∗}[P − Q] = ϕ(P) − ϕ(Q) = ([nP] − D_{P} ) − ([nQ] − D_{Q}) = [nP − nQ] − (D_{P} − D_{Q}), so D_{P} = D_{Q}.

Now assume ϕ(P) = [nP] − D for all P ∈ C(k̄). Then ϕ^{σ}(P) = (ϕ(P^{σ^{−1}}))^{σ} = ([nP^{σ^{−1}}] − D)^{σ} = [nP] − D^{σ}, so ϕ = ϕ^{σ} for all σ ∈ Gal(k̄/k) if and only if D = D^{σ} for all σ ∈ Gal(k̄/k), i.e., if and only if the class D is k-rational. □

Let us write

C [figure: (n)] J_{C}  (II.1)

for a k-morphism with the property that the induced map ϕ_{∗} on jacobians is multiplication-by-n. Given two diagrams

C [figure: (n)] J_{C} and C′ [figure: (n)] J_{C′},

a morphism between them is a curve morphism ϕ: C [figure] C′ fitting into a commutative diagram

[figure: C; C′; ϕ; J_{C′}.; (n); J_{C}; (n); ϕ_{∗}]

By the proof of II.3.1, each (II.1) is equal to j_{D} for a unique D ∈ Pic^{n}(C)(k), whence the two horizontal arrows in the diagram are of the form j_{D} and j_{D′}. Commutativity of the diagram is easily seen to correspond to the condition ϕ_{∗}D = D′.

Thus there is an isomorphism between the category of quasi-elliptic curves and the category of morphisms of the form (II.1), which goes as follows: to a quasi-elliptic curve (C, D) we associate the morphism j_{D}, while to a morphism (II.1), which must equal j_{D} for a unique D ∈ Pic^{n}(C)(k), we associate the quasi-elliptic curve (C, D).

De-emphasizing the divisor class by thinking of a quasi-elliptic curve as being a morphism (II.1) is analogous to de-emphasizing the origin by thinking of an elliptic curve as being a curve together with an isomorphism to its jacobian.

II.3a. Relationship with n-coverings of an elliptic curve

Let (E, O) be a fixed elliptic k-curve. Associated with E is the Kummer sequence

0 [figure] \frac{E(k)}{nE(k)} [figure] H^{1}(G_{k}, E[n](k̄)) [figure] H^{1}(G_{k}, E(k̄))[n] [figure] 0.  (∗)

Elements of the Weil–Châtelet group WC(E) := H^{1}(G_{k}, E(k̄)) are classically described as equivalence classes of torsors (= principal homogeneous spaces) over E: a k-curve C together with a simply transitive action of E on C. Elements of WC(E) may instead be described (see [AKM^{+}01, §4]) as equivalence classes of pairs

(C, J_{C} [figure: ∼] E),

where J_{C} [figure: ∼] E is a k-isomorphism of elliptic curves.

Elements of H^{1}(G_{k}, E[n](k̄)) are classically described (following [Cas60, §1] and [Cas62, §2]) as equivalence classes of n-coverings: diagrams

\xymatrix{C \ar[d]_{∼} \ar[r] & E \\ E \ar[ur]_{[n]}}  (∗∗)

in which C and the morphism C [figure] E are each defined over k, but the isomorphism C [figure: ∼] E need not be defined over k. Elements of H^{1}(G_{k}, E[n](k̄)) may instead be described as equivalence classes of pairs

(C [figure: (n)] J_{C}, J_{C} [figure: ∼] E),

where C [figure: (n)] J_{C} is a quasi-elliptic k-curve (in the sense of §II.3), and J_{C} [figure: ∼] E is a k-isomorphism

of elliptic curves; two pairs (C [figure: (n)] J_{C}, J_{C} [figure: ∼] E) and (C′ [figure: (n)] J_{C′}, J_{C′} [figure: ∼] E) are equivalent when there is a k-morphism ϕ: C [figure] C′ so that the following diagram commutes:

[figure: C; C′; ϕ; J_{C′}; (n); E.; ∼; E; J_{C}; (n); ϕ_{∗}; ∼]

We now establish this alternate description. Given a pair (C [figure: (n)] J_{C}, J_{C} [figure: ∼] E), we immediately obtain the corresponding classical n-covering (∗∗). On the other hand, given (∗∗), C [figure] E induces the pullback morphism J_{E} [figure] J_{C} whose kernel is J_{E}[n]. Thus J_{E}/J_{E}[n] ≅ J_{C}, but also J_{E}/J_{E}[n] ≅ J_{E} ≅ E, whence there is a canonical k-isomorphism J_{C} [figure: ∼] E.

With these descriptions, the first map in (∗) sends a point P ∈ E(k) to the map Q [figure] nQ + P, or more precisely,

\begin{aligned}[P] [figure] & (E [figure: (n)] J_{E}, J_{E} [figure: ∼] E) \\ & (Q [figure] [nQ + P − (n + 1)O], [P − O] [figure] P ), \end{aligned}

while the second map in (∗) is simply the forgetful map

(C [figure: (n)] J_{C}, J_{C} [figure: ∼] E) [figure] (C, J_{C} [figure: ∼] E).

II.4. Mapping a quasi-elliptic curve to projective space

Consider a quasi-elliptic curve (C, D), of degree n, for which the divisor class D admits a k-rational representative D. Such a quasi-elliptic curve, together with the additional data of the choice of representative, is denoted simply (C, D). Associated with (C, D) is the k-morphism ι_{D} of degree n to the canonical projective space bundle arising from D, which is given on k̄-valued points by

\begin{matrix}ι_{D}: C(k̄) & [figure] P(H^{0}(C, O(D)))(k̄), \\ P & [figure] \{ s ∈ H^{0}(Ck̄, O(D)) : s(P) = 0 \}. \end{matrix}

Remark. For V any k-vector space, the k̄-valued points of the projective space bundle P(V) are simply the hyperplanes in the k̄-vector space V ⊗_{k} k̄. (See appendix B for details on projective space bundles.) The map ι_{D} takes a point to the hyperplane of sections that vanish at that point, but one must interpret this correctly: O(D) is a subsheaf of the constant sheaf induced by the function field, so it is common to think of a section of O(D) as a rational function f; the section vanishes at P precisely when the order of vanishing of f at P is at least one higher than what is allowed/required by the coefficient of P in D.

The map ι_{D} is an embedding precisely for n ≥ 3, because that is the precise condition for D to be very ample (see [Har77, IV.3.3.3]). We will study such embedded curves in chapter III. For now, however, we make no assumption on n.

By choosing a basis B = { x_{0}, …, x_{n−1} } for H^{0}(C, O(D)), our map becomes

\begin{matrix}ι_{B}: C(k̄) & [figure] P_{k}^{n−1}(k̄), \\ P & [figure] [x_{0}(P) : …: x_{n−1}(P)]. \end{matrix}

The image of ι_{B} is non-degenerate (meaning: it does not lie in a hyperplane), because otherwise we would contradict the linear independence of the basis elements. Each choice of basis leads to such a map, and two different choices lead to two maps that differ by an automorphism of P_{k}^{n−1}, namely by the element of PGL_{n}(k) induced by the change-of-basis matrix. Replacing D with D′ ∈ [D]_{k}, and choosing a basis B′ for H^{0}(C, O(D′)), we obtain a map ι_{B′}: C [figure] P_{k}^{n−1} as before, and it, too, differs from the original ι_{B} by an automorphism of P_{k}^{n−1}. (To see this, note that O(D) and O(D′) are isomorphic, whence by B.3b we have P(H^{0}(C, O(D))) ≅ P(H^{0}(C, O(D′))). Now apply [Har77, II.7.1.1].)

In short, given a quasi-elliptic curve (C, D) with the property that D admits a k-rational representative but without the data of a choice of such representative, there is a maximal family F of k-morphisms ι: C [figure] P_{k}^{n−1} of degree n with non-degenerate image such that any two members of F differ by a k-rational automorphism of P_{k}^{n−1}. Given the pair (C, F), we recover the pair (C, D) by letting D be the class of divisors of hyperplane sections (= pullbacks of hyperplanes) for any (and all) maps in F.

II.4a. Extending a morphism between quasi-elliptic curves to a morphism between projective spaces

Isomorphisms between quasi-elliptic curves induce compatible morphisms between the projective spaces to which they map. More precisely, if (C, D) is a quasi-elliptic curve with a k-rational divisor of degree n, and if (C′, D′) is a quasi-elliptic curve with a k-rational divisor of the same degree n, and if we choose a basis B for H^{0}(C, O(D)) and a basis B′ for H^{0}(C′, O(D′)), and if we have a morphism ϕ: C [figure] C′ with ϕ_{∗}D ∼ D′, then we can ask about the existence of a map in the indicated position:

[figure: C; P_{k}^{n−1}; ι_{B}; C′; ϕ; P_{k}^{n−1}; ι_{B′}; ?]  (∗)

Proposition II.4.1. If ϕ is an isomorphism, then the map in question in (∗) exists (and is an automorphism of P_{k}^{n−1}).

Proof. We have ϕ^{∗}D′ ∼ D. The desired map is obtained by composing the maps in the following commutative diagram:

[figure: C; P(H^{0}(C, O(D))); P(H^{0}(C, O(ϕ^{∗}D′))); ∼; (by B.3b); P(H^{0}(C, ϕ^{∗}O(D′))); ∼; (by B.3b); P(H^{0}(C′, O(D′))).; (by B.3a); C; C; C′; ∼; ϕ]

□

It is natural to wonder, when ϕ is not an isomorphism, whether the map in question in (∗) exists. We now show this is in general not the case.

Assume it exists. By pulling back O(1) from the lower right corner of (∗) to the upper left corner in two different ways, we obtain O(ℓD) ≅ ϕ^{∗}O(D′) for some ℓ ∈ Z, which forces ℓ = deg(ϕ), and we have obtained a simple necessary condition:

ϕ^{∗}D′ ∼ deg(ϕ) · D.

Example. The necessary condition need not hold. Let (C, O) be an elliptic curve with a k-rational point P of order 2. Let Φ be the subgroup { O, P }. The quotient map C [figure] C/Φ may be viewed as a degree 2 morphism between quasi-elliptic curves of degree 3:

ϕ: (C, 3O) [figure] (C/Φ, 3O/Φ).

Then ϕ^{∗}(3O/Φ) = 3O + 3P, while deg(ϕ) · 3O = 6O, but [3]O ⊕ [3]P ≠ [6]O, whence 3O + 3P ≁ 6O. Therefore, if C and C/Φ are embedded in P_{k}^{2} via 3O and 3O/Φ, then ϕ does not extend to a morphism P_{k}^{2} [figure] P_{k}^{2}.

CHAPTER III

Projective quasi-elliptic curves C [figure: n]P_{k}^{n−1}

In this chapter, we study arithmo-geometric aspects of k-curves of genus 1 that are embedded as non-degenerate degree n curves in P_{k}^{n−1}, where k is a perfect field. Obviously n ≥ 3. Such a curve C will be denoted

C [figure: n] P_{k}^{n−1}.  (III.1)

Remark. Together with the additional data of a choice of k-valued point on J_{C}[n], and the assumption char(k) ∤ n, (III.1) is called an “n-prepared curve” in [O’N01]. When k = C, (III.1) is called an “elliptic normal curve”.

Let D be the class of hyperplane sections on C. Then (III.1) gives us a quasi-elliptic curve (C, D) of degree at least 3 and such that D admits a k-rational representative. Call such a quasielliptic curve projective. Conversely, given a k-curve C of genus 1 and a divisor class D on C of degree n ≥ 3 that admits a k-rational representative D, a choice of basis for H^{0}(C, O(D)) gives an embedding to projective space (cf. §II.4), recovering the description (III.1). In summary, “projective quasi-elliptic curve of degree n” means essentially the same thing as “non-degenerate degree n curve in P_{k}^{n−1} of genus 1”.

Note that each k-curve C of genus 1 occurs as a projective quasi-elliptic curve: simply define a k-rational divisor on C by taking the Galois orbit of any k̄-valued point on C.

III.1. Attempting to simplify the description

In this section, we answer the question: are any of the words in “non-degenerate degree n curve in P_{k}^{n−1} of genus 1” superfluous? After all, we are familiar with the fact: a degree 3 smooth curve in P_{k}^{2} necessarily has genus 1 and necessarily is non-degenerate.

Proposition III.1.1. If X ⊂ P_{k}^{n−1} is a non-degenerate curve, then deg(X) ≥ n − 1.

Proof. Choosing n − 1 distinct points in X(k̄), there exists a hyperplane H ⊂ Pn−1 ; k̄ containing those points. If deg(X) < n − 1, then Xk̄ is contained in H. □

Proposition. If X ⊂ P_{k}^{n−1} is a non-degenerate curve, and D is a hyperplane section, then ℓ(D) ≥ n.

Proof. The dimension ℓ(D) is the same for every hyperplane section, so let D be the section determined by X_{0} = 0. Then 1, X_{1}/X_{0}, …, X_{n−1}/X_{0} lie in H^{0}(X, O(D)), and they are linearly independent since X is non-degenerate. Thus ℓ(D) ≥ n. □

Corollary. If C ⊂ P_{k}^{n−1} is a non-degenerate curve of genus 1, then deg(C) ≥ n.

Proof. By Riemann–Roch (see §A.1), we have ℓ(D) = deg(D) for any divisor of positive degree. Now let D be any hyperplane section. Then deg(D) = deg(C). □

Example. We show that a non-degenerate curve in P_{k}^{n−1} of genus 1 can have degree strictly larger than n. In [Har77, §IV.3] it is shown that every projective k-curve can be embedded in P_{k}^{3}. The procedure goes like this: if the curve already lies in P_{k}^{2} or P_{k}^{3}, we’re done; otherwise, we repeatedly project from points off the curve to decrease the dimension. We can always get as far as P_{k}^{3} without

III.3. The saturated homogeneous ideal of C 25

introducing singularities. Thus we can start with a projective quasi-elliptic curve C [figure: 5]P_{k}^{4} of degree 5 in P_{k}^{4}, and then project it to P_{k}^{3}, giving us a non-degenerate degree 5 curve in P_{k}^{3} of genus 1.

Example. We show that a degree n curve in P_{k}^{n−1} of genus 1 can be degenerate. Take the nondegenerate degree 5 curve in P_{k}^{3} of genus 1 from the previous example, and inject it into P_{k}^{4}. This gives a degenerate degree 5 curve in P_{k}^{4} of genus 1.

Example. We show that a non-degenerate degree n curve in P_{k}^{n−1} can have genus ≠ 1. Consider a rational normal curve (generalization of the twisted cubic) of degree n in P_{k}^{n} and project it to P_{k}^{n−1}. It has genus 0.

In summary, no words in “non-degenerate degree n curve in P_{k}^{n−1} of genus 1” are superfluous.

III.2. Basic facts about C [figure: n] P_{k}^{n−1}

Let C [figure: n] P_{k}^{n−1} be a projective quasi-elliptic curve.

Proposition III.2.1. Given n − 1 points P_{1}, …, P_{n−1} ∈ C(k̄), they lie in the support of a unique hyperplane section.

Proof. By §A.1d, there is a unique Q ∈ C(k̄) so that P_{1} + … + P_{n−1} + Q ∈ D, where D is the divisor class comprising hyperplane sections on C. □

Corollary III.2.2. Given n − 1 distinct points P_{1}, …, P_{n−1} ∈ C(k̄), their linear span in P_{k}^{n−1} is a hyperplane.

Proof. If their linear span had codimension ≥ 2, then P_{1}, …, P_{n−1} would be contained in more than one hyperplane, contradicting III.2.1. □

Corollary III.2.3. Any n − 1 or fewer distinct points in C(k̄) are independent.

Corollary III.2.4. The linear span of any n − 2 or fewer points in C(k̄) misses the rest of C.

Proof. Let ℓ be the number of distinct points in the given collection. By III.2.3, they span a Pℓ ; k̄. If that Pℓ ; k̄ were to meet C in another point, then, again by III.2.3, the points would have to instead

span a Pℓ+1 ; k̄. □

III.3. The saturated homogeneous ideal of C

From the scheme viewpoint, our projective quasi-elliptic curve C is a closed subscheme of

P_{k}^{n−1} = Proj(S), where S = k[X_{0}, …, X_{n−1}],

and thus, by [Har77, II.5.16], we have

C = Proj(S/I),

where I is a homogeneous ideal in S. However, if I and I′ are homogeneous ideals of S such that the rings S/I and S/I′ are not isomorphic, it can nonetheless be the case (see [Har77, Ex. II.2.14c]) that

Proj(S/I) ≅ Proj(S/I′).  (III.2)

Fortunately, by [Har77, Ex. II.5.10], we have (III.2) if and only if I and I′ have the same saturation. Recall that the saturation of an ideal I is

Ī = { s ∈ S : for each i, there exists n_{i} so that X_{i}^{n_{i}} · s ∈ I }.

We always have Proj(S/I) ≅ Proj(S/Ī), and the correspondence between closed subschemes of P_{k}^{n−1} and saturated homogeneous ideals in S is one-to-one. Whenever we speak of the ideal of C, we always mean the unique saturated homogeneous ideal in S that defines C.

26 III. Projective quasi-elliptic curves C [figure: n] P_{k}^{n−1}

More generally, by [Har77, II.5.9], closed subschemes (of any scheme, not just projective space) correspond to quasi-coherent sheaves of ideals. Let I_{C} denote the ideal sheaf of C on P_{k}^{n−1}. It is defined by the exactness of the following sequence of O_{P_{k}^{n−1}}-modules:

0 [figure] I_{C} [figure] O_{P_{k}^{n−1}}  [figure] ι_{∗}O_{C} [figure] 0.  (III.3)

The relationship between I_{C} and the ideal of C is given by [Har77, Ex. II.5.10c]:

I = ⨁_{m∈Z}H^{0}(P_{k}^{n−1}, I_{C}(m)).  (III.4)

It is worth recording that the (saturated homogeneous) ideal of Ck̄ is simply

Ik̄ = I ⊗_{k} k̄.

Certainly Ik̄ is an ideal of Ck̄, and one can easily check that it is saturated; alternatively, start with the description

Ik̄ := ⨁_{m∈Z}H^{0}(Pn−1 ; k̄, I_{Ck̄} (m)),

and apply [Har77, III.9.3], which tells us

H^{0}(P_{k}^{n−1}, I_{C}(m)) ⊗_{k} k̄ = H^{0}(Pn−1 ; k̄, I_{Ck̄} (m)), for all m ∈ Z,

whence we recover Ik̄ = I ⊗_{k} k̄.

III.4. Projective normality

By twisting (III.3), we obtain, for each m ∈ Z, the following exact sequence of O_{P_{k}^{n−1}}-modules:

0 [figure] I_{C}(m) [figure] O_{P_{k}^{n−1}}(m) [figure] (ι_{∗}O_{C})(m) [figure] 0.  (III.5)

Taking global sections, we obtain

0 [figure] H^{0}(P_{k}^{n−1}, I_{C}(m)) [figure] H^{0}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(m)) [figure] H^{0}(C, O_{C}(mD))

[figure] H^{1}(P_{k}^{n−1}, I_{C}(m)) [figure] H^{1}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(m)) [figure] H^{1}(C, O_{C}(mD)) [figure] …, (III.6)

where D is any choice of k-rational hyperplane section on C. (We used H^{0}(P_{k}^{n−1}, (ι_{∗}O_{C})(m)) = H^{0}(P_{k}^{n−1}, ι_{∗}(O_{C}(mD))) = H^{0}(C, O_{C}(mD)), which follow from [Har77, Ex. III.4.1, II.5.12c].)

It is of interest to determine for which m the map

H^{0}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(m)) [figure] H^{0}(C, O_{C}(mD))  (III.7)

from (III.6) is surjective. This gives us information about the dimension of the first term in (III.6), which, by (III.4), is information about the mth graded piece of the ideal I of C.

For m < 0, surjectivity of (III.7) follows from ℓ(mD) = 0 (see [Har77, IV.1.2]). For m = 0, surjectivity follows from the fact that C is connected. For m = 1, surjectivity is sometimes called linear normality, and holds because we can view (III.1) as the embedding associated with a choice of basis for H^{0}(C, O_{C}(D)), whence the homogeneous coordinates on P_{k}^{n−1} pull back to that basis. We now establish surjectivity for all m, which is sometimes called projective normality (cf. [Har77, Ex. II.5.14]).

Theorem III.4.1. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. For each m ∈ Z, the short exact sequence (III.5) remains exact upon taking global sections. In other words, the map (III.7) is surjective.

Proof. Since H^{1}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(m)) = 0 (see [Har77, III.5.1]), the surjectivity of (III.7) is equivalent, by (III.6), to

H^{1}(P_{k}^{n−1}, I_{C}(m)) = 0, for all m ∈ Z.

The cases m < 2 were established preceding the theorem statement. Now we mimic the proof in [Hul86].

By [Har77, III.9.3], H^{1}(P_{k}^{n−1}, I_{C}(m)) ⊗_{k} k̄ ≅ H^{1}(Pn−1 ; k̄, I_{Ck̄} (m)), whence we may assume k = k̄. By Bertini (see [Har77, II.8.18]), there exists a hyperplane H so that the scheme D = C ∩H comprises n distinct points, call them P_{1}, …, P_{n}.

\xymatrix{D \ar[r]^{s} \ar[d] & H \ar[d]^{j} \\ C \ar[r]^{ι} & P_{k}^{n−1}}

We will now consider various sheaves of O_{P_{k}^{n−1}}-modules on P_{k}^{n−1}, which fit together into the following diagram:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & I_{C}(m) \ar[d] \ar[r] & O_{P_{k}^{n−1}}(m) \ar[d] \ar[r] & (ι_{∗}O_{C})(m) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & I_{C}(m + 1) \ar[d] \ar[r] & O_{P_{k}^{n−1}}(m + 1) \ar[d] \ar[r] & (ι_{∗}O_{C})(m + 1) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & (j_{∗}I_{D/H})(m + 1) \ar[d] \ar[r] & (j_{∗}O_{H})(m + 1) \ar[d] \ar[r] & (j_{∗}s_{∗}O_{D})(m + 1) \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

Multiplication by the linear form ℓ ∈ H^{0}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(1)) that defines H gives the exact sequence

0 [figure] I_{C}(m) [figure: ℓ] I_{C}(m + 1) [figure] (j_{∗}I_{D/H})(m + 1) [figure] 0,

where I_{D/H} denotes the ideal sheaf of D on H. Taking global sections, we obtain

… [figure] H^{1}(P_{k}^{n−1}, I_{C}(m))

[figure] H^{1}(P_{k}^{n−1}, I_{C}(m + 1)) [figure] H^{1}(P_{k}^{n−1}, (j_{∗}I_{D/H})(m + 1)) [figure] …,

and the theorem is reduced, by induction on m ≥ 1, to establishing

H^{1}(P_{k}^{n−1}, (j_{∗}I_{D/H})(m + 1)) = 0, for m ≥ 1.  (∗)

If we now start with the exact sequence of sheaves of O_{H}-modules

0 [figure] I_{D/H} [figure] O_{H} [figure] s_{∗}O_{D} [figure] 0,

we can apply j_{∗} and then twist m + 1 times to obtain

0 [figure] (j_{∗}I_{D/H})(m + 1) [figure] (j_{∗}O_{H})(m + 1) [figure] (j_{∗}s_{∗}O_{D})(m + 1) [figure] 0.

Note that j_{∗} is exact here because j is a closed immersion (see [Har77, III.3.7, III.8.1]). Taking global sections gives

… [figure] H^{0}(H, O_{H}(m + 1)) [figure] H^{0}(P_{k}^{n−1}, (j_{∗}s_{∗}O_{D})(m + 1))

[figure] H^{1}(P_{k}^{n−1}, (j_{∗}I_{D/H})(m + 1)) [figure] H^{1}(H, O_{H}(m + 1)) [figure] ….

We now claim H^{1}(H, O_{H}(m+1)) vanishes for m ≥ 1; in other words, since H ≅ P_{k}^{n−2}, we are claiming H^{1}(P_{k}^{n−2}, O_{P_{k}^{n−2}}(m+1)) = 0 for m ≥ 1. For n ≥ 4, the claim follows immediately from [Har77, III.5.1b]. For n = 3, the claim follows from the existence of a perfect pairing (see [Har77, III.5.1d])

H^{0}(P_{k}^{1}, O_{P_{k}^{1}}(−m−3))×H^{1}(P_{k}^{1}, O_{P_{k}^{1}}(m+1)) [figure] k and the fact that the first term in this pairing vanishes when m ≥ 1 (see [Har77, II.5.13]). Having established the claim, we see that (∗) is equivalent to showing the map

H^{0}(H, O_{H}(m + 1)) [figure] H^{0}(P_{k}^{n−1}, (j_{∗}s_{∗}O_{D})(m + 1))

to be surjective for m ≥ 1. Since the codomain is simply the direct sum of one-dimensional skyscraper sheaves supported on the P_{i}, we are done if we can exhibit an (m + 1)-form on H that vanishes at all P_{i} except, say, P_{1}. Since some projective coordinate does not vanish at P_{1}, we are reduced to finding a quadratic form on H that vanishes at all P_{i} except P_{1}.

By lemma III.2.4, the points P_{3}, …, P_{n} span a linear space in P_{k}^{n−1} that misses the rest of C, and thus they define a hyperplane inside H that misses P_{1} and P_{2}. Reasoning instead with P_{2}, …, P_{n−1}, we obtain a hyperplane that misses P_{1} and P_{n}. The product of the associated linear forms gives a quadratic form on H that vanishes at all P_{i} except P_{1}. □

III.5. The Hilbert function of C

To describe I explicitly for particular values of n, we will start by determining the dimension of each graded piece. Set

d_{I}(s) := dim_{k} I_{s} = dim_{k} H^{0}(P_{k}^{n−1}, I_{C}(s)).

By III.4.1, taking global sections of the exact sequence

0 [figure] I_{C}(s) [figure] O_{P_{k}^{n−1}}(s) [figure] (i_{∗}O_{C})(s) [figure] 0

gives the exact sequence

0 [figure] H^{0}(P_{k}^{n−1}, I_{C}(s)) [figure] H^{0}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(s)) [figure] H^{0}(C, O(sD)) [figure] 0,

whence

d_{I}(s) = dim_{k} H^{0}(P_{k}^{n−1}, O_{P_{k}^{n−1}}(s)) − ℓ(sD).

By counting monomials of degree s, and by Riemann–Roch, we obtain

d_{I}(s) = \binom{n − 1 + s}{n − 1} − ns.

Recall that the Hilbert function of C is ϕ_{C}(s) := dim_{k}(S/I)_{s}. By what we determined above, we immediately obtain:

ϕ_{C}(s) = dim_{k} S_{s} − dim_{k} I_{s}

= \binom{n − 1 + s}{n − 1} − d_{I}(s)

= ns.

For sufficiently large s, the Hilbert function agrees with a polynomial function (see [Har77, I.7.5]), called the Hilbert polynomial of C. But ϕ_{C}(s) is itself polynomial in s, whence ϕ_{C}(s) is both the Hilbert function and the Hilbert polynomial. (Since C is a degree n curve of genus 1, we can independently see, by [Har77, I.7.6, Ex. I.7.2], that its Hilbert polynomial is P_{C}(s) = ns.) We have proved:

Theorem III.5.1. On a non-degenerate degree n curve in P_{k}^{n−1} of genus 1, the Hilbert polynomial and Hilbert function are the same, namely ϕ_{C}(s) = P_{C}(s) = ns.

III.6. C is a complete intersection for n = 3 and n = 4 29

III.6. C is a complete intersection for n = 3 and n = 4

We ask: for which n is a projective quasi-elliptic curve C a complete intersection?

Recall, for X a closed subscheme of P_{k}^{n−1}, each generating set for the ideal of X has cardinality ≥ n − 1 − dim(X); if there exists a generating set of cardinality = n − 1 dim(X), then X is said to be a complete intersection, in which case the degree of X is the product of the degrees of the n − 1 − dim(X) generators (see [Har77, I.7.7]).

Our curve (III.1), whose degree n is ≥ 3, is a complete intersection if and only if its ideal can be generated by n − 2 elements. Since no hyperplane contains C, each generator has degree ≥ 2, whence, to be a complete intersection, C must have degree ≥ 2^{n−2}. From 2^{n−2} ≤ n we conclude n ≤ 4.

Theorem III.6.1. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Then C is a complete intersection if and only if n = 3 or n = 4.

Proof. We already saw that n = 3 or n = 4 is necessary. Let n = 3. From our calculation in §III.5, we have

d_{I}(s) := dim_{k} I_{s} = \binom{s + 2}{s} − 3s = \frac{(s − 2)(s − 1)}{2}.

In particular, d_{I}(3) = 1, so there is a cubic form F ∈ I. Since ⟨F ⟩ ⊆ I, we obtain ⟨F ⟩ = I by showing dim_{k}⟨F ⟩_{s} = dim_{k} I_{s} for each degree s. But the dimension of ⟨F ⟩ in degree s is the same as the dimension of S = k[x, y, z] in degree s − 3. Since dim_{k} S_{s} = \binom{s+2}{2}, we have dim_{k}⟨F ⟩_{s} = \binom{s−1}{2}, which is just d_{I}(s).

Now let n = 4. Then

d_{I}(s) = \binom{s + 3}{s} − 4s = \frac{(s − 1)(s^{2} + 7s − 6)}{6}.

In particular, d_{I}(2) = 2, so there are two linearly independent quadratic forms Q_{1}, Q_{2} ∈ I. As before, we will show dim_{k} I_{s} = dim_{k}⟨Q_{1}, Q_{2}⟩_{s}, and thus obtain I = ⟨Q_{1}, Q_{2}⟩. Since there are relations between Q_{1} and Q_{2}, we have to work a bit harder to determine dim_{k}⟨Q_{1}, Q_{2}⟩_{s}. Let N be the free module S ⊕ S, where S = k[w, x, y, z]. Consider the Koszul complex for (Q_{1}, Q_{2}) ∈ N (cf. [Eis95, §17.2]),

0 [figure] S [figure] N [figure] ⋀^{2} N [figure] 0,

where the maps are just wedge-multiplication on the left by the element (Q_{1}, Q_{2}). In terms of the basis e_{1} = (1, 0) and e_{2} = (0, 1) for N, and the basis e_{1} ∧ e_{2} for ⋀^{2} N, the complex is

[figure: 0; S; S^{2}; ^{\binom{Q_{1}}{Q_{2}}}; S; (−Q_{2},Q_{1}); 0.]  (∗)

By [Eis95, 17.5], the complex (∗) is exact everywhere except at the right since Q_{1}, Q_{2} compose a regular sequence; indeed, since C is not contained in a plane, Q_{1} is irreducible whence ⟨Q_{1}⟩ is prime, and since Q_{2} is not a multiple of Q_{1}, it is nonzero in the integral domain S/⟨Q_{1}⟩. We thus obtain the exact sequence

[figure: 0; S; S^{2}; ^{\binom{Q_{1}}{Q_{2}}}; ⟨Q_{1}, Q_{2}⟩; (−Q_{2},Q_{1}); 0.]  (∗∗)

From this we can determine dim_{k}⟨Q_{1}, Q_{2}⟩_{s}. Since dim_{k} S_{s} = \binom{s+3}{3}, the dimension of the middle term of (∗∗) in degree s is 2\binom{s+3}{3}. The map \binom{Q_{1}}{Q_{2}} shifts degrees by 2, whence the dimension of the cokernel in degree s is 2\binom{s+3}{3} − \binom{(s−2)+3}{3}. But the quotient map (−Q_{2},Q_{1}) again shifts degrees by 2, so

dim_{k}⟨Q_{1}, Q_{2}⟩_{s} = 2\binom{(s − 2) + 3}{3} − \binom{(s − 4) + 3}{3},

which is the same as d_{I}(s). □

Remark. For n = 4, an alternate argument appears at the end of §18.2 in [Eis95].

III.7. Cutting C out by quadrics when n ≥ 4 and char(k) ≠ 2

We ask: is C cut out scheme-theoretically by quadrics (equivalently, is I generated by quadratic forms)? If so, how many quadrics are needed? In the previous section, we saw that C is not cut out by quadrics when n = 3, while it is cut out by a pair of quadrics when n = 4.

Proposition III.7.1. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Then each minimal generating set for I contains n(n − 3)/2 quadratic forms.

Proof. From our work in §III.5, we have

dim_{k}\{ quadratic forms in I \} = d_{I}(2) = dim_{k} H^{0}(P_{k}^{n−1}, I_{C}(2)) = \frac{n(n − 3)}{2}.

Since there are no linear forms in I, each generating set for I must contain n(n − 3)/2 linearly independent quadratic forms, whence each minimal generating set contains precisely n(n − 3)/2 quadratic forms (and possibly also forms of higher degree—indeed, such is the case when n = 3). □

Theorem III.7.2. Assume char(k) ≠ 2. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Then C is cut out scheme-theoretically by quadrics if and only if n ≥ 4, in which case each minimal generating set for I comprises n(n − 3)/2 elements, all of them quadratic forms.

Proof. In light of III.7.1, it remains to show, when n ≥ 4 and char(k) ≠ 2, that I is generated by quadratic forms—we do this below, in III.7.4. □

We recall (cf. [Har77, Ex. 5.12] and [Har95, Lec. 3, Lec. 22]) basic facts about quadrics in P_{k}^{n−1} when char(k) ≠ 2. Thinking of the variables (X_{0}, …, X_{n−1}) as a column vector X, each quadratic form Q(X) ∈ k[X_{0}, …, X_{n−1}] can be written in the form Q(X) = X^{t}AX, where A is an n × n symmetric matrix with entries in k. The rank of the quadric defined by Q is, by definition, simply the rank of the matrix A. A rank 1 quadric is a hyperplane of multiplicity 2, while a rank 2 quadric is a union of two distinct hyperplanes. A quadric is irreducible if and only if its rank r satisfies r ≥ 3, and a quadric is smooth if and only if r is maximal: r = n. For 2 ≤ r ≤ n − 1, the quadric may be described as a cone, with vertex a P_{k}^{n−1−r}, over a smooth (rank r) quadric in P_{k}^{r−1}. Given any two distinct quadrics Q_{A} and Q_{B}, with (scheme-theoretic) intersection X = Q_{A} ∩ Q_{B}, each member of the pencil

λ_{0}Q_{A} + λ_{1}Q_{B}, where [λ_{0} : λ_{1}] ∈ P_{k}^{1}(k̄),  (III.8)

also contains X. The pencil (III.8) may be recovered by any two distinct members of that pencil. The singular members correspond to the roots [λ_{0} : λ_{1}] ∈ P_{k}^{1}(k̄) of the equation

det(λ_{0}A + λ_{1}B) = 0,  (III.9)

where A and B are the n × n symmetric matrices corresponding to Q_{A} and Q_{B}.

We return now to our curve C. Since C is non-degenerate, each quadric containing C has rank ≥ 3, and we will show that Ck̄ is in fact the (scheme-theoretic) intersection of the rank 3 quadrics containing it. (We had to go to k̄ for the following reason: since the roots of (III.9) need not be k-rational, the singular quadrics in Pn−1 ; k̄ containing Ck̄ need not be defined over k.)

For n = 4, we already know from §III.6 that C = Q_{A} ∩ Q_{B}, where Q_{A} and Q_{B} are any two distinct quadrics containing C, and we know from III.7.1 that every quadric containing C lies in the pencil (III.8). (Both facts hold even when char(k) = 2, but henceforth we assume char(k) ≠ 2.)

We will see later (in §III.13) that there exists a linear change-of-coordinates, defined over k̄, so that the group H′_{4} ⊂ GL_{4}(k̄) of order 64 generated by

M = [\begin{matrix}1 \\ & i \\ & & −1 \\ & & & −i \end{matrix}] and N = [\begin{matrix}& 1 \\ & & 1 \\ & & & 1 \\ 1 \end{matrix}]

III.7. Cutting C out by quadrics when n ≥ 4 and char(k) ≠ 2 31

preserves Ck̄ in its action on P3 k̄, where i^{2} = −1. (We use the notation H′_{4} because, in §III.12, we reserve “H_{4}” to mean the order 64 subgroup of SL_{4}(k̄) that has the same image in PGL_{4}(k̄) as H′_{4}.) We assume the change-of-coordinates to have been applied.

Since H′_{4} preserves Ck̄, it also preserves the set (III.8) of all quadrics that contain Ck̄. As discussed in [Hul86, §III.2], the H′_{4}-module H^{0}(P3 k̄, O(2)) can be expressed as a direct sum of irreducible submodules in the form

H^{0}(P3 k̄, O(2)) = (V_{1} ⊕ V_{2}) ⊕ V_{3} ⊕ V_{4} ⊕ V_{5},

where

\begin{aligned}V_{1} & = ⟨x_{0}^{2} + x_{2}^{2}, x_{1}^{2} + x_{3}^{2}⟩, \\ V_{2} & = ⟨x_{1}x_{3}, x_{0}x_{2}⟩, \\ V_{3} & = ⟨x_{0}^{2} − x_{2}^{2}, x_{1}^{2} − x_{3}^{2}⟩, \\ V_{4} & = ⟨x_{0}x_{1} + x_{2}x_{3}, x_{1}x_{2} + x_{0}x_{3}⟩, \\ V_{5} & = ⟨x_{0}x_{1} − x_{2}x_{3}, x_{1}x_{2} − x_{0}x_{3}⟩. \end{aligned}

One can check that V_{1} ≅ V_{2} as H′_{4}-modules, while no other two of the direct summands are isomorphic. Therefore, either our pencil (III.8) lies in V_{1} ⊕ V_{2}, or it is equal to one of V_{3}, V_{4}, V_{5}. It cannot be V_{3} since Ck̄ does not lie in a plane, while it can be neither V_{4} nor V_{5} since those pencils define singular curves. Thus, our pencil lies in V_{1} ⊕ V_{2}. As an H′_{4}-module, it must be isomorphic to V_{1} and V_{2}, whence it must have a k̄-vector space basis on which H′_{4} acts in the same manner as it does on the bases for V_{1} and V_{2} exhibited above; in short, our pencil must have a k̄-vector space basis of the form

⟨a(x_{0}^{2} + x_{2}^{2}) + b(x_{1}x_{3}), a(x_{1}^{2} + x_{3}^{2}) + b(x_{0}x_{2})⟩.

Some pairs of values for a, b cannot occur, but we need not identify them all: certainly (a, b) ≠ (0, 0), and it is easy to check that (III.9) has at least two distinct roots. In summary, Ck̄ is an intersection of a pair of quadrics of rank 3, whence it is also the intersection of all rank 3 quadrics containing it.

Theorem III.7.3. Assume k = k̄ and char(k) ≠ 2. For n ≥ 4, each non-degenerate degree n curve in P_{k}^{n−1} of genus 1 is cut out scheme-theoretically by the rank 3 quadrics containing it; that is, the rank 3 quadratic forms in I compose a generating set for I.

Proof. The proof for k = C in [Hul86, §IV.1] goes through unaltered. For the convenience of the reader, we repeat it here. It proceeds by induction on n. The case n = 4 was explained preceding the theorem.

Assume n ≥ 5. We shall first show that C is a set-theoretic intersection of rank 3 quadrics. To do this, let P ∈ P_{k}^{n−1}(k) \ C(k) be an arbitrary point not lying on C. We can choose a point P_{0} ∈ C(k) such that the line P P_{0} is neither a secant nor a tangent of C. (Otherwise, projection from P would map C onto a non-degenerate curve C′ ⊂ P_{k}^{n−2} of degree ≤ n/2, so by III.1.1 we would have n/2 ≥ n − 2, which is never the case for n ≥ 5.) Projection from P_{0} maps C onto a non-degenerate degree n − 1 curve C′ ⊂ P_{k}^{n−2} of genus 1, and the image P′ of P does not lie on C′. By the induction hypothesis, there is a rank 3 quadric Q′ through C′ that does not contain P′. Let Q be the cone over Q′ with vertex P_{0}. Then C ⊂ Q but P ∉ Q(k).

It remains to show that the intersection is scheme-theoretic; equivalently, we must show that the rank 3 quadrics separate tangents. Let P ∈ C(k) be an arbitrary point on C and let L be a line through P that is not tangent to C at P. Next choose a point P_{0} ∈ C(k) that does not lie on L. Projecting from P_{0} we get a non-degenerate degree n − 1 curve C′ in P_{k}^{n−2} of genus 1. Let P′ and L′ be the images of P and L. By the induction hypothesis, there is a rank 3 quadric Q′ through C′ such that Q′ and the line L′ intersect transversally at P′. The cone Q over Q′ with vertex P_{0} therefore intersects the line L at P transversally. □

The following corollary to the theorem completes the proof of III.7.2.

Corollary III.7.4. Assume char(k) ≠ 2. For n ≥ 4, each non-degenerate degree n curve in P_{k}^{n−1} of genus 1 is cut out scheme-theoretically by quadrics; that is, the ideal I of C is generated by quadratic forms. (Moreover, in any generating set for I, the subset of quadratic forms is itself a generating set.)

Proof. By the theorem, certainly I ⊗_{k} k̄ is generated by quadratic forms. Let { p_{1}, …, p_{r} } be a finite set of homogeneous forms that generate I. Any element of I ⊗_{k} k̄ can also be expressed in terms of the p_{i}. Certainly none of the p_{i} is linear, so by degree considerations, elements of a quadratic generatoring set for I ⊗_{k} k̄ can be expressed in terms of those p_{i} that are quadratic, whence the quadratic p_{i} themselves compose a quadratic generating set for I ⊗_{k} k̄. Multiplying them by all possible powers of the variables leads to a generating set for I ⊗_{k} k̄ as a vector space over k̄. Thus there is a k̄-basis for I ⊗_{k} k̄ of vectors that are defined over k, whence, by linear descent (see [Sil99, II.5.8.1]), the same vectors compose a k-basis for I. Therefore, the quadratic p_{i} compose a quadratic generating set for I. □

III.8. Classifying linear automorphisms of C [figure: n] P_{k}^{n−1}

Let C [figure: n] P_{k}^{n−1} be a projective quasi-elliptic curve of degree n, and let D be the class of hyperplane sections on C.

The automorphism group of Pn−1 ; k̄ is PGL_{n}(k̄). Since elements of PGL_{n}(k̄) send hyperplanes in Pn−1 ; k̄ to hyperplanes, if such an element also preserves Ck̄, then it will induce an automorphism of (Ck̄, D). Automorphisms of Ck̄ that are induced by automorphisms of the ambient projective space are called linear automorphisms of Ck̄.

It turns out that all automorphisms of (Ck̄, D) are linear. We saw in II.4.1 that each automorphism of (Ck̄, D) extends to an automorphism of the ambient Pn−1 ; k̄. Furthermore, the extension is unique: since each point in Pn−1 ; k̄(k̄) is an intersection of hyperplanes in Pn−1 ; k̄, an element of PGL_{n}(k̄) is determined by its behavior on hyperplanes, which are in one-to-one correspondence with hyperplane sections on Ck̄; thus, if two elements of PGL_{n}(k̄) move a given hyperplane to two different places, then they move the associated hyperplane section to two different places, and thus cannot induce the same automorphism of Ck̄. We have shown:

Aut((Ck̄, D)) = { linear automorphisms of Ck̄ } ≅ { ϕ ∈ PGL_{n}(k̄): ϕ(Ck̄) = Ck̄ }.

Note that the isomorphism above is Gal(k̄/k)-equivariant. In other words: any group of matrix classes in PGL_{n}(k̄) that acts on C necessarily does so in a faithful and Galois equivariant manner. By II.2.7, the largest such group has structure

{ ϕ ∈ PGL_{n}(k̄) : ϕ(Ck̄) = Ck̄ } ≅ J_{C}[n](k̄) ⋊ Aut((Ck̄, O)),

where O ∈ C(k̄) is a point of hyperosculation.

Proposition III.8.1. In the above identification, the elements of PGL_{n}(k̄) that act fixpoint-free on C correspond precisely to J_{C}[n](k̄). Thus, when char(k) ∤ n, there are precisely n^{2} elements of PGL_{n}(k̄) that act fixpoint-free on C.

Proof. The first part is a special case of A.4.3. The second part follows from #J_{C}[n](k̄) = n^{2} when char(k) ∤ n. □

Corollary III.8.2. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Let J_{n} ⊂ PGL_{n}(k̄) be the subgroup of elements that preserve C and act fixpoint-free on C. Then J_{n} is a faithful and Galois equivariant representation of the action of J_{C}[n](k̄) on C.

III.9. Concerning the [−1]-automorphisms of C [figure: n] P_{k}^{n−1}

This section is about those linear automorphisms of Ck̄ that are of order 2 and have a fixpoint in C(k̄).

III.9. Concerning the [−1]-automorphisms of C [figure: n] P_{k}^{n−1} 33

Proposition III.9.1. Let ϕ be an order 2 curve automorphism of Ck̄ that has a fixpoint P ∈ C(k̄). Then ϕ is the unique order 2 automorphism of the elliptic curve (Ck̄, P), more commonly known as [−1]. Thus the fixpoints of ϕ are the points of 2-torsion on (Ck̄, P); furthermore, if Q is such a point, then ϕ is also the [−1]-automorphism on the elliptic curve (Ck̄, Q).

Proof. See A.4.8. □

Let O ∈ C(k̄) be a point of hyperosculation. By [−1] we mean the unique order 2 automorphism of the elliptic curve (Ck̄, O). In the previous section, we identified the linear automorphisms of Ck̄ with J_{C}[n](k̄) ⋊ Aut((Ck̄, O)).

Proposition III.9.2. Inside J_{C}[n](k̄) ⋊ Aut((Ck̄, O)), the order 2 linear automorphism of Ck̄ with a fixpoint are precisely the elements of the form τ ∘ [−1], where τ ∈ J_{C}[n](k̄).

Proof. Follows immediately from A.4.9. □

It is easy to verify (cf. the proof of A.4.9) that τ_{Q⊕Q} ∘ [−1] is the unique order 2 automorphism of Ck̄ that fixes Q ∈ C(k̄), where the group law is on the elliptic curve (Ck̄, O), and τ_{Q⊕Q} is the translation-by-(Q ⊕ Q) map. We therefore see a relationship between points of hyperosculation and fixpoints of order 2 linear automorphisms, but due to the presence of Q ⊕ Q, that relationship depends on the parity of n:

• Assume n is odd. For P ∈ C(k̄) a point of hyperosculation, there is a unique point of hyperosculation Q ∈ C(k̄) so that [2]Q = P. Therefore, for each linear automorphism of order 2 with a fixpoint in C(k̄), precisely one of its fixpoints is a point of hyperosculation.

• Assume n is even. For P ∈ C(k̄) a point of hyperosculation, either all the solutions of [2]Q = P are points of hyperosculation, or none of them are. (The two cases are determined by whether P is a point of (n/2)-torsion.) Therefore, for each linear automorphism of order 2 with a fixpoint in C(k̄), either all of its fixpoints are points of hyperosculation, or none of them are.

In either case, we see: each point of hyperosculation occurs as the fixpoint of a unique order 2 linear automorphism.

III.9a. Procedure for finding the points of hyperosculation

By the work in §III.8, we may identify the order 2 linear automorphisms of C that have a fixpoint with the elements of PGL_{n}(k̄) that: act on C, have order 2, and admit a fixpoint. In fact, as the following proposition shows, if we find one of them, and if we have found all the elements of PGL_{n}(k̄) that act fixpoint-free on C, then this gives us all the order 2 elements with fixpoints.

Proposition III.9.3. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Let T̄ ∈ PGL_{n}(k̄) be an element of order 2 that preserves C and has at least one fixpoint on C. The remaining such automorphisms have the form M̄ ⊗ T̄, where M̄ ∈ PGL_{n}(k̄) acts on C without fixpoints.

Proof. If one of the fixpoints of T̄ is a point of hyperosculation, then this proposition is just a repeat of III.9.2. In fact, by III.9.2, we know that T̄ = τ ∘ [−1] for some τ ∈ J_{C}[n](k̄). It is clear that each M̄ ∘ T̄ also has that form. □

To find the points of hyperosculation, we first find all the order 2 elements of PGL_{n}(k̄) that preserve C and have a fixpoint on C. The points of hyperosculation lie among the finitely many fixpoints. For each fixpoint, the osculating hyperplane can be determined by brute force: parameterize all hyperplanes that go through the given point, and then impose the conditions that maximize the intersection multiplicity of the hyperplane with C. The unique solution is the osculating hyperplane. If the hyperplane does not meet C elsewhere, then the point in question is a point of hyperosculation.

Remark. When n is odd, a simpler way for finding the points of hyperosculation is given in III.11.8.

III.10. The commutator pairing is the Weil pairing

Let J_{n} ⊂ PGL_{n}(k̄) denote the subgroup of elements that act fixpoint-free on C. In the previous section, we saw that there is a canonical isomorphism of Galois modules

J_{n} ≅ J_{C}[n](k̄).  (III.10)

We will establish that the “commutator pairing” (defined below) on J_{n} corresponds, under (III.10), to the inverse of the Weil pairing on J_{C}[n](k̄).

III.10a. Commutator pairings on abelian projective linear groups

Consider an abelian subgroup G ⊂ PGL_{n}(k̄). For matrix classes [A], [B] ∈ G, the usual commutator

[[A], [B]] := [A] [B] [A]^{−1}[B]^{−1}

is of course trivial. But if we first lift the classes [A], [B] to representing matrices A, B ∈ GL_{n}(k̄), then the commutator [A, B] might be nontrivial, and all we can say for certain is that [A, B] lies in k̄^{×}. It is easy to check that [A, B] is independent of the choice of representing matrices. Thus we obtain a well-defined commutator pairing

\begin{matrix}G × G & [figure] k̄^{×}, \\ ([A], [B]) & [figure] A B A^{−1}B^{−1}. \end{matrix}

The commutator pairing is immediately seen to be alternating (whence also skew-symmetric), and can be easily verified to be bilinear. If G is Galois stable, then the pairing is a map between Galois modules, and it is easily seen to be equivariant with respect to that action.

Remark. The commutator pairing could easily be degenerate, indeed trivial, as is seen by considering the matrices

[\begin{matrix}1 & 0 & 0 \\ 0 & ζ_{3} & 0 \\ 0 & 0 & ζ_{3}^{2} \end{matrix}] and [\begin{matrix}1 & 0 & 0 \\ 0 & ζ_{3} & 0 \\ 0 & 0 & ζ_{3} \end{matrix}] ,

which generate a Z/3Z × Z/3Z in PGL_{3}(Q̄).

III.10b. The Weil pairing on an elliptic curve (E, O)

Let (E, O) be an elliptic curve, defined over k, and let n be such that char(k) ∤ n. Then as a group, E[n](k̄) ≅ Z/nZ × Z/nZ, although as a Galois module the structure can be more complicated. The well-known Weil pairing is usually defined by one of the procedures for computing it, such as the one in [Sil99, III.8]. We will instead work with the description given below in III.10.1, and connect that with the commutator pairing below in III.10.3.

Theorem. Assume char(k) ∤ n. The Weil pairing

e_{n}: E[n](k̄) × E[n](k̄) [figure] μ_{n}(k̄)

has the following properties:

• bilinear

• alternating, whence skew-symmetric;

• perfect: E[n](k̄) ≅ Hom(E[n](k̄), μ_{n}(k̄)), whence non-degenerate and surjective;

• Galois equivariant;

• compatible: e_{mn}(P, Q) = e_{n}(mP, Q);

• admits dual isogenies as adjoints: e_{n}(P, ϕQ) = e_{n}(ϕ̂P, Q).

Proof. All properties except “perfect” are established in [Sil99, III.8.1, III.8.1.1, III.8.2]. As for “perfect”, observe that non-degeneracy gives an injection E[n](k̄) [figure] Hom(E[n](k̄), μ_{n}(k̄)), which must be an isomorphism since the two sets have the same cardinality. □

E[n](k̄) admits other pairings with all the properties listed above: simply post-compose the Weil pairing with an automorphism of μ_{n}(k̄). Among these pairings, it isn’t so obvious which of them should be considered the true Weil pairing. For example, the pairing as defined in [Sil99, III.8] is actually the inverse of the pairing as described in [Mum70]. The latter agrees with the description of the pairing in terms of function-on-divisor evaluation given in [Sil99, Ex. 3.16], which we now recall.

Theorem III.10.1. Assume char(k) ∤ n. Let (E, O) be an elliptic curve, and let e_{n} denote the Weil pairing on E[n](k̄). Given P, Q ∈ E[n](k̄), we may compute e_{n}(P, Q) as follows. Choose divisors D_{P} and D_{Q} with disjoint support so that D_{P} ∼ P − O and D_{Q} ∼ Q − O. Choose rational functions f_{P} and f_{Q} so that (f_{P}) = nD_{P} and (f_{Q}) = nD_{Q}. Then

e_{n}(P, Q) = \frac{f_{P} (D_{Q})}{f_{Q}(D_{P})}.

Proof. See [Sil99, Ex. 3.16]. □

Proposition III.10.2. Two points P, Q ∈ E[n](k̄) are generators for E[n](k̄) if and only if e_{n}(P, Q) is a generator for μ_{n}(k̄), i.e., if and only if e_{n}(P, Q) is a primitive nth root of unity.

Proof. Let the pair (P, Q) generate. Since the Weil pairing is surjective, e_{n}(P, Q) is a primitive nth root of unity. Each pair of elements in E[n](k̄) can be written in the form (aP + bQ, cP + dQ), with a, b, c, d ∈ Z/nZ. Such a pair generates E[n](k̄) if and only if there is a group automorphism of E[n](k̄) carrying (P, Q) to that pair; in other words, the matrix

[\begin{matrix}a & c \\ b & d \end{matrix}]

must lie in GL_{2}(Z/nZ), which is equivalent to the condition: ad − bc is a unit in Z/nZ. On the other hand, e_{n}(aP + bQ, cP + dQ) is easily seen to be e_{n}(P, Q)^{ad−bc}, which is a primitive nth root of unity under precisely the same condition: ad − bc is a unit in Z/nZ. □

III.10c. The commutator pairing on J_{n} is the Weil pairing

We return to the subgroup J_{n} ⊂ PGL_{n}(k̄) and the canonical isomorphism (III.10). Since J_{n} is abelian, it admits the commutator pairing

J_{n} × J_{n} [figure] k̄^{×}.

If char(k) ∤ n, then J_{C}[n](k̄) admits the Weil pairing

J_{C}[n](k̄) × J_{C}[n](k̄) [figure] μ_{n}(k̄).

We ask: in light of (III.10), what is the relationship between the two pairings?

The following theorem is sketched in [O’N01, Thm. 2.5], and appears for the case k = C in [Hul86]. We give here an elementary proof.

Theorem III.10.3. Assume char(k) ∤ n. Let C be a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. Then, under the isomorphism (III.10), the commutator pairing on J_{n} corresponds to the inverse of the Weil pairing on J_{C}[n](k̄).

Remark. If one defines the Weil pairing as done in [Sil99, III.8], then one must eliminate “the inverse of” from the theorem statement. (Cf. discussion preceding III.10.1.

Proof. Let P̄_{0}, …, P̄_{n^{2}−1} ∈ C(k̄) be the points of hyperosculation on C. For each P̄_{i}, there exists a hyperplane H̄_{i} meeting C to order n at P̄_{i}, and an element M̄_{i} ∈ J_{n} whose action on C (given by matrix multiplication) is translation-by-P̄_{i} on the elliptic curve (C, P̄_{0}):

M̄_{i}P̄_{0} = P̄_{i}.

We wish to compute e_{n}(M̄_{i}, M̄_{j}). By III.10.1, we need disjoint divisors D_{i} and D_{j} linearly equivalent, respectively, to P̄_{i} − P̄_{0} and P̄_{j} − P̄_{0}. We set

\begin{aligned}D_{i} & := P̄_{i} − P̄_{0}, \\ D_{j} & := (P̄_{j} ⊕ P̄_{ℓ}) − P̄_{ℓ}, \end{aligned}

where ⊕ denotes addition on the elliptic curve (C, P̄_{0}), and ℓ is such that

P̄_{ℓ} ∉ { P̄_{0}, P̄_{i}, ⊖P̄_{j}, P̄_{i} ⊖ P̄_{j} }.

This is the precise condition for the divisors to be disjoint, and such ℓ exists because n ≥ 3.

In what follows, the following notation will be useful: P̄_{i⊕j} is the point P̄_{i} ⊕ P̄_{j}, H̄_{i⊕j} is the hyperplane associated to P̄_{i⊕j}, and M̄_{i⊕j} is the matrix class giving translation-by-P̄_{i⊕j}.

Now we lift everything to row vectors, column vectors, and matrices: choose row vectors H_{i} representing H̄_{i}, column vectors P_{i} representing P̄_{i}, and matrices M_{i} representing M̄_{i}. By P_{i⊕j}, H_{i⊕j}, and M_{i⊕j} we mean the row vector, column vector, and matrix chosen to represent P̄_{i⊕j}, H̄_{i⊕j}, and M̄_{i⊕j}.

Observe that H_{i}/H_{0} is a function on C(k̄) with divisor nD_{i}, while H_{j⊕ℓ}/H_{ℓ} is a function on C(k̄) with divisor nD_{j}. By III.10.1, we have

e_{n}(M̄_{i}, M̄_{j}) = \frac{(H_{i}/H_{0})(D_{j})}{(H_{j⊕ℓ}/H_{ℓ})(D_{i})} = \frac{H_{i}P_{j⊕ℓ}}{H_{i}P_{ℓ}} \frac{H_{0}P_{ℓ}}{H_{0}P_{j⊕ℓ}} \frac{H_{j⊕ℓ}P_{0}}{H_{j⊕ℓ}P_{i}} \frac{H_{ℓ}P_{i}}{H_{ℓ}P_{0}} .

Since each vector appears once in the numerator and once in the denominator, everything we have done is independent of our choice of lifts. Since M̄_{i} corresponds to translation-by-P̄_{i}, we see that M_{i}P_{0} represents P̄_{i}, but M_{i}P_{0} need not equal P_{i}: they differ by a scalar. We can nonetheless apply the substitution P_{i} [figure] M_{i}P_{0}, so long as we do so simultaneously in the numerator and in the denominator. We can similarly apply H_{i} [figure] H_{0}M_{i}^{−1}, since H_{0}M_{i}^{−1} represents H̄_{i}. Applying these types of substitutions gives us

e_{n}(M̄_{i}, M̄_{j}) = \frac{H_{0}M_{i}^{−1}M_{j⊕ℓ}P_{0}}{H_{0}M_{i}^{−1}M_{ℓ}P_{0}} \frac{H_{0}M_{ℓ}P_{0}}{H_{0}M_{j⊕ℓ}P_{0}} \frac{H_{0}M_{j⊕ℓ}^{−1}P_{0}}{H_{0}M_{j⊕ℓ}^{−1}M_{i}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{i}P_{0}}{H_{0}M_{ℓ}^{−1}P_{0}} .

Since M_{i⊕j} and M_{i}M_{j} each represent M̄_{i}M̄_{j}, we apply M_{i⊕j} [figure] M_{i}M_{j} to obtain

e_{n}(M̄_{i}, M̄_{j}) = \frac{H_{0}M_{i}^{−1}M_{j}M_{ℓ}P_{0}}{H_{0}M_{i}^{−1}M_{ℓ}P_{0}} \frac{H_{0}M_{ℓ}P_{0}}{H_{0}M_{j}M_{ℓ}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{j}^{−1}P_{0}}{H_{0}M_{ℓ}^{−1}M_{j}^{−1}M_{i}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{i}P_{0}}{H_{0}M_{ℓ}^{−1}P_{0}} .

By §III.9, the [−1] automorphism of the elliptic curve (C, P̄_{0}) lies in PGL_{n}(k̄). Let T be a matrix representing this automorphism. Then T P_{0} and P_{0} differ by a scalar, similarly H_{0}T and H_{0} differ by a scalar, and the same can be said for M_{i}T and T M_{i}^{−1}. Paying attention that each of the following moves in the numerator is matched by a corresponding move in the denominator, the pesky scalars will never make an appearance. We insert T appropriately, and then walk it to the left one step at a time until it disappears again:

\frac{H_{0}M_{i}^{−1}M_{j}M_{ℓ}P_{0}}{H_{0}M_{i}^{−1}M_{ℓ}P_{0}} \frac{H_{0}M_{ℓ}P_{0}}{H_{0}M_{j}M_{ℓ}P_{0}} = \frac{H_{0}M_{i}^{−1}M_{j}M_{ℓ}T P_{0}}{H_{0}M_{i}^{−1}M_{ℓ}T P_{0}} \frac{H_{0}M_{ℓ}T P_{0}}{H_{0}M_{j}M_{ℓ}T P_{0}}

= \frac{H_{0}M_{i}^{−1}M_{j}T M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}^{−1}T M_{ℓ}^{−1}P_{0}} \frac{H_{0}T M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}T M_{ℓ}^{−1}P_{0}} = \frac{H_{0}M_{i}^{−1}T M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}^{−1}T M_{ℓ}^{−1}P_{0}} \frac{H_{0}T M_{ℓ}^{−1}P_{0}}{H_{0}T M_{j}^{−1}M_{ℓ}^{−1}P_{0}}

= \frac{H_{0}T M_{i}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}T M_{i}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{ℓ}^{−1}P_{0}} = \frac{H_{0}M_{i}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{ℓ}^{−1}P_{0}} .

Inserting this equation already leads to some cancellation:

e_{n}(M̄_{i}, M̄_{j}) = \frac{H_{0}M_{i}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{j}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{i}P_{0}}{H_{0}M_{ℓ}^{−1}M_{j}^{−1}M_{i}P_{0}} .

III.11. The configuration of hyperplanes fixed by J_{n} 37

Now we would like to swap adjacent pairs of matrices. Although we have not yet shown matrix commutators to be the Weil pairing, those commutators nonetheless are scalars, so we must merely be careful to match each swap above and below.

e_{n}(M̄_{i}, M̄_{j}) = \frac{H_{0}M_{i}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{ℓ}^{−1}M_{i}P_{0}}{H_{0}M_{j}^{−1}M_{ℓ}^{−1}M_{i}P_{0}}

= \frac{H_{0}M_{i}M_{j}^{−1}M_{ℓ}^{−1}P_{0}}{H_{0}M_{i}M_{ℓ}^{−1}P_{0}} \frac{H_{0}M_{i}M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{i}M_{ℓ}^{−1}P_{0}}

= [M_{i}, M_{j}^{−1}]\frac{H_{0}M_{j}^{−1}M_{i}M_{ℓ}^{−1}P_{0}}{H_{0}M_{j}^{−1}M_{i}M_{ℓ}^{−1}P_{0}} = [M_{i}, M_{j}^{−1}] = [M_{i}, M_{j}]^{−1}.

The final equality of commutators holds because the commutator is a scalar. □

III.11. The configuration of hyperplanes fixed by J_{n}

We return to the subgroup of PGL_{n}(k̄) that preserves C (see §III.8), especially the subgroup J_{n}— corresponding to J_{C}[n](k̄)—that preserves C and has no fixpoints on C. The reader may easily verify the following facts about fixpoints and fixed hyperplanes:

• Let M̄ ∈ PGL_{n}(k̄), and let M ∈ GL_{n}(k̄) represent M̄. A fixpoint of M̄ corresponds to an eigenvector of M.

• M̄ fixes a linear subspace of Pn−1 ; k̄ pointwise if and only if the corresponding linear subspace of An ; k̄ is an eigenspace of M.

• For a row vector ℓ̄ of homogeneous coordinates, let H_{ℓ} := { P̄ ∈ P_{k}^{n−1}(k̄) : ℓ̄· P̄ = 0 } be the corresponding hyperplane. The action of PGL_{n}(k̄) on points is P̄ [figure] M̄P̄, but its action on hyperplanes is ℓ̄ [figure] ℓ̄M̄^{−1}:

M̄H_{ℓ} = { M̄P̄ : ℓ̄· P̄ = 0 } = { P̄ : ℓ̄M̄^{−1}P̄ = 0 } = H_{ℓM̄^{−1}}.

• M̄ fixes a hyperplane H_{ℓ} if and only if ℓ is an eigenvector of M^{t}. Concerning the structure of things related to J_{n}, there are often three cases to consider:

• #J_{n} = n^{2}, which happens if and only if char(k) ∤ n, in which case we have J_{n} ≅ Z/nZ × Z/nZ (as Z-modules);

• #J_{n} = n^{2}/p^{e} where p = char(k) and p^{e} | n but p^{e+1} ∤ n, which can happen when char(k) | n, in which case we have J_{n} ≅ Z/nZ × Z/(n/p^{e})Z (as Z-modules);

• #J_{n} = n^{2}/p^{2e}, which is the other possibility when char(k) | n, in which case we have

J_{n} ≅ Z/(n/p^{e})Z × Z/(n/p^{e})Z (as Z-modules).

As we will see, we can often make a general statement—without restricting to a particular case above—in terms of an element of J_{n} of order n. Implicitly, of course, such a statement applies only to the first two cases given above.

Proposition III.11.1. Let M̄ ∈ PGL_{n}(k̄) be an element that preserves C. For each hyperplane H fixed by M̄, the corresponding hyperplane section H ∩ C is a union of orbits under the action of M̄ on C.

Proof. Obvious. □

Corollary III.11.2. Let M̄ ∈ J_{n} have order n. Then each hyperplane H fixed by M̄ meets C transversally: H ∩ C comprises n distinct points.

Proof. Each orbit of M̄ on C comprises n distinct points. On the other hand, H ∩ C comprises at most n points. Now apply the proposition. □

Proposition III.11.3. Assume that n is odd. Let M̄ ∈ J_{n} have order n. The orbit of Q ∈ C(k̄) is a hyperplane section if and only if Q is a point of hyperosculation.

Proof. Fix a point of hyperosculation O ∈ C(k̄) to obtain a group law. Then points of n-torsion coincide with points of hyperosculation. The orbit of Q is a hyperplane section if and only if Q+M̄Q+ M̄^{2}Q + … + M̄^{n−1}Q ∼ nO. But M̄ is just translation-by-P for some point of hyperosculation P. Thus the condition is Q + (Q ⊕ P) + (Q ⊕ [2]P) + … + (Q ⊕ [n − 1]P) ∼ nO, which is equivalent to [n]Q ⊕ [n(n − 1)/2]P = O, which reduces to: Q is a point of hyperosculation. □

To obtain a version of the proposition for n even, we need the notion of a dyadic packet. By the isomorphism C/J_{C}[n] ≅ J_{C} from II.2.4, the n-torsion packets on C correspond to the k̄-valued points on J_{C}, and the hyperosculation packet on C corresponds to the origin on J_{C}. We will call an n-torsion packet on C dyadic if it corresponds to a point of 2-torsion on J_{C}.

Proposition III.11.4. Fix a point of hyperosculation O ∈ C(k̄) to obtain a group law on Ck̄. The multiplication-by-n map C [figure] C gives a one-to-one correspondence between the dyadic packets and the 2-torsion on (Ck̄, O). Thus the dyadic packets partition the (2n)-torsion on (Ck̄, O); furthermore, this partition is independent of the choice of O.

Proof. It is easy to check that the group law on J_{C} is induced by the group law on (Ck̄, O), where O is any choice of point of hyperosculation. The preimages under multiplication-by-n of the 2-torsion on (Ck̄, O) are easily seen to be n-torsion packets, with the property that multiplying them by 2 sends them into the hyperosculation packet; in short, they are the 2-torsion on the quotient J_{C}. □

Proposition III.11.5. Assume that n is even. Let M̄ ∈ J_{n} have order n. Let a point of hyperosculation O ∈ C(k̄) define a group law, so that M̄ corresponds to translation-by-P, whence M̄^{n/2} is translation by [n/2]P, which is a point of order 2 on (Ck̄, O). Then the orbit of Q ∈ C(k̄) is a hyperplane section if and only if [n]Q = [n/2]P, i.e., if and only if Q lies in the dyadic packet corresponding to [n/2]P.

Proof. By the proof of III.11.3, the orbit of Q is a hyperplane section if and only if [n]Q ⊕ [n(n − 1)/2]P = O, which reduces to: [n]Q = [n/2]P. □

Combining III.11.3 and III.11.5, we obtain: if M̄ ∈ J_{n} has order n, then the orbit of Q ∈ C(k̄) is a hyperplane section if and only if Q lies in a certain n-torsion packet that depends only on M̄. From this we obtain the following result.

Corollary III.11.6. Let M̄ ∈ J_{n} have order n, and let M ∈ GL_{n}(k̄) represent M̄.

• Each eigenvalue of M has a 1-dimensional eigenspace.

• M has #J_{n}/n distinct eigenvalues.

Proof. There are only finitely many points in an n-torsion packet, so there can be only finitely many hyperplanes fixed by M̄. If an eigenspace of M were 2-dimensional or larger, then there would be infinitely many hyperplanes fixed by M̄.

The n-torsion packet has cardinality #J_{n}. That set is partitioned into collections of size n by the hyperplanes fixed by M̄. The fixed hyperplanes correspond to the eigenspaces of M, which in turn correspond to the eigenvalues. □

Corollary III.11.7. Let M̄ ∈ J_{n} have order n, and let M ∈ GL_{n}(k̄) represent M̄.

• If char(k) ∤ n (i.e., #J_{n} = n^{2}), then M has n distinct eigenvalues, whence there are precisely n distinct fixpoints of M̄ (not on C), and precisely n distinct fixed hyperplanes. The fixed hyperplanes intersect C in the n^{2} points of either the hyperosculation packet (when n is odd) or a non-hyperosculation dyadic packet (when n is even).

• If char(k) | n (i.e., #J_{n} = n^{2}/p^{e} where p = char(k) and p^{e} | n but p^{e+1} ∤ n), then M has n/p^{e} unique eigenvalues, whence there are precisely n/p^{e} distinct fixpoints of M̄ (not on C), and precisely n/p^{e} distinct fixed hyperplanes. The fixed hyperplanes intersect C in the n/p^{e} points of either the hyperosculation packet (when n is odd) or a non-hyperosculation dyadic packet (when n is even).

• Special case: if n = p is prime and char(k) = p (i.e., #J_{n} = p), then M has a unique eigenvalue (algebraic multiplicity n, but geometric multiplicity 1), thus a unique fixpoint (not on C), and a unique fixed hyperplane. The fixed hyperplane intersects C in the p points of either the hyperosculation packet (when p is odd) or the unique non-hyperosculation dyadic packet (when p = 2).

Remark III.11.8. When n is odd, this gives an easy procedure to find the points of hyperosculation on C: find an element of PGL_{n}(k̄) that preserves C, has no fixpoints, and has order n; next, find its fixed hyperplane(s); finally, compute their intersection with C. (When char(k) | n and furthermore #J_{n} = n^{2}/p^{2e} where p = char(k) and p^{e} | n but p^{e+1} ∤ n, then this procedure can’t be used. Instead, one can follow the procedure in §III.9a.)

III.11a. Hyperplane configurations when char(k) ∤ n

For the rest of this section, we will assume char(k) ∤ n, whence J_{n} ≅ Z/nZ × Z/nZ (as Z-modules). Our goal is to describe the configuration of hyperplanes fixed by generators for J_{n}.

We already know, by III.11.6, that (a representative for) M̄ ∈ J_{n} of order n has distinct eigenvalues. Now we can say a bit more.

Proposition III.11.9. Assume char(k) ∤ n. Let M̄ ∈ J_{n} be an element of order n, and let M ∈ GL_{n}(k̄) be any matrix representating M̄. If λ ∈ k̄^{×} is an eigenvalue of M, then also ζλ is an eigenvalue of M for each ζ ∈ μ_{n}(k̄). In other words, M has distinct eigenvalues, and they compose a coset of μ_{n}(k̄) inside k̄^{×}.

Proof. Since M̄ has order n, there exists N̄ ∈ J_{n} so that M̄ and N̄ are generators. Let N be any lift of N̄ to GL_{n}(k̄). By III.10.2 and III.10.3, MNM^{−1}N^{−1} is a primitive nth root of unity, which we’ll call ζ_{n}. Now let λ ∈ k̄^{×} be an eigenvalue of M, and let x be a corresponding eigenvector. Then

Mx = λx, and M(Nx) = MNx = ζ_{n}NMx = ζ_{n}λNx = (ζ_{n}λ)(Nx),

so ζ_{n}λ is an eigenvalue of M (with corresponding eigenvector Nx). □

Proposition III.11.10. Assume char(k) ∤ n. Let M̄, N̄ ∈ J_{n} be generators. Then M̄ cyclically permutes the hyperplane sections preserved by N̄, and vice versa.

Proof. In the proof of III.11.9, we saw that if x is an eigenvector of M, also Nx is an eigenvector of M. Thus N̄ cyclically permutes the fixpoints of M̄, whence also N̄ cyclically permutes the hyperplanes preserved by M̄. □

Theorem III.11.11. Assume char(k) ∤ n. Further assume that n is odd. If M̄, N̄ generate J_{n}, then there is a one-to-one correspondence

\left\{{hyperplane sections \atop preserved by M̄}\right\} × \left\{{hyperplane sections \atop preserved by N̄}\right\} [figure] [figure] \{ points of hyperosculation \}
(H, H′) [figure] H ∩ H′,

which is equivariant with respect to the action of J_{n}; thus, if we fix a point of hyperosculation as origin to obtain a group law, this correspondence is an isomorphism of J_{n}-modules.

Proof. Obvious from the preceding material. □

Instead of intersecting hyperplane sections to obtain points in C(k̄), we could instead intersect the actual hyperplanes to obtain points in a Grassmannian. When n is even, the hyperplanes fixed by M cover a non-hyperosculation dyadic packet on C, while the hyperplanes fixed by N cover a different non-hyperosculation dyadic packet on C. There is again an intersection pairing taking values in a Grassmannian, but not one taking values in C(k̄). (In other words, when n is even, the (n−3)-dimensional linear space obtained by intersecting a hyperplane fixed by M with a hyperplane fixed by N does not meet C.)

III.12. Lifting J_{n} ⊂ PGL_{n}(k̄) to H_{n} ⊂ GL_{n}(k̄)

We now consider the problem of lifting the subgroup J_{n} ⊂ PGL_{n}(k̄), which corresponds to J_{C}[n](k̄), from PGL_{n}(k̄) to GL_{n}(k̄). In other words, we seek a finite subgroup H_{n} ⊂ GL_{n}(k̄) so that, under the canonical projection GL_{n} [figure] PGL_{n}, the image of H_{n} is J_{n}. We furthermore require H_{n} to be a Gal(k̄/k)-module.

Of course we would like H_{n} to be as small as possible; in fact, we would be delighted if H_{n} could be isomorphic, as a Galois module, to J_{n}. Unfortunately, as we will see below, this is too much to ask in general.

Remark. One reason for wanting H_{n} to be Gal(k̄/k)-stable is that it may then be viewed as the set of k̄-valued points of a finite group scheme defined over k. This will be relevant in chapter IV, where we will be interested in computing a quotient by J_{n}, which we will express in terms of invariants of H_{n}. This also explains why we would like H_{n} to be as small as possible: the computational expense for finding the elements of H_{n} and then working out its invariant theory is likely to increase with the size of H_{n}.

We always have available the following canonical lift: we define H_{n} by taking the preimage of J_{n} under the canonical projection SL_{n} [figure] PGL_{n}:

[figure: 1; μ_{n}(k̄); SL_{n}(k̄); PGL_{n}(k̄); 1; 1; μ_{n}(k̄); H_{n}; J_{n}; 1]

In other words:

H_{n} := { A ∈ GL_{n}(k̄) : Ā ∈ J_{n} and det(A) = 1 }.

Note #H_{n} = #μ_{n}(k̄) · #J_{n}. (Thus, when char(k) ∤ n, we have #H_{n} = n^{3}.) Since the determinant 1 condition is preserved when matrices are moved by elements of Gal(k̄/k), the canonical lift H_{n} is Gal(k̄/k)-stable. We will see below that H_{n} is always a central extension of J_{n} by μ_{n}(k̄).

III.12a. Case: char(k) ∤ n (i.e., #J_{n} = n^{2})

We saw that taking commutators of arbitrary lifts of elements in J_{n} corresponds to the Weil pairing (see III.10.3). Since the Weil pairing is surjective, any lift of J_{n} to GL_{n}(k̄) must contain the constant matrices 1, ζ_{n}, ζ_{n}^{2}, …, ζ_{n}^{n−1}. Therefore, any lift of J_{n} has cardinality ≥ n^{3}, while the canonical lift H_{n} has cardinality = n^{3}.

Now that we know lifts of order n^{3} exist, we can classify them all. Let H′_{n} be a fixed but unknown lift of J_{n} to GL_{n}(k̄) of order n^{3}. For M̄ ∈ J_{n}, let M ∈ H_{n} and M′ ∈ H′_{n} denote a choice of representatives for M̄. Since both H_{n} and H′_{n} admit only elements of μ_{n}(k̄) as scalar matrices, we have (M′)^{n} = αM^{n} for some α ∈ μ_{n}(k̄), and α is independent of our choices. Thus we obtain a well-defined homomorphism

\begin{matrix}ϕ: J_{n} & [figure] μ_{n}(k̄) \\ M̄ & [figure] (M′)^{n}M^{−n} \end{matrix}

On the other hand, if ϕ ∈ Hom(J_{n}, μ_{n}(k̄)), then we can define a lift H′_{n} as follows: take generators M̄, N̄ for J_{n}, choose representatives M, N ∈ H_{n}, set M′ = \sqrt[n]{ϕ(M̄)}M and N′ = \sqrt[n]{ϕ(N̄)}N for some choice of nth roots, and let H′_{n} be the group of order n^{3} generated by M′, N′. It is easy to check that we have established a one-to-one correspondence between Hom(J_{n}, μ_{n}(k̄)) and lifts of J_{n} to GL_{n}(k̄) of order n^{3}; in fact, the correspondence respects the Galois action, in the following sense: if ϕ is the homomorphism associated to H′_{n}, then ϕ^{σ} is the homomorphism associated to (H′_{n})^{σ}. Thus the lifts that are Galois stable correspond to the elements of Hom(J_{n}, μ_{n}(k̄)) that are Galois invariant. The canonical lift H_{n} corresponds to the trivial homomorphism.

Proposition. Assume char(k) ∤ n. There exist precisely n^{2} lifts of J_{n} to GL_{n}(k̄) of order n^{3}.

III.13. Schrödinger-like representations of H_{n} when char(k) ∤ n 41

Proof. The cardinality of Hom(J_{n}, μ_{n}(k̄)) is n^{2}. □

If we fix generators M̄, N̄ for J_{n}, and a generator ζ_{n} for μ_{n}(k̄), and choose representatives M′, N′ ∈ H′_{n}, then each element of H′_{n} has a unique expression in the form

ζ_{n}^{a}(M′)^{b}(N′)^{c}, where a, b, c ∈ Z/nZ.  (III.11)

Proposition. Assume char(k) ∤ n. Each of the lifts H′_{n} is a central extension of J_{n} by μ_{n}(k̄).

Proof. Using commutators, we can express a product of elements in the form (III.11) in the same form. It is easy to check that an element lies in the center if and only if b ≡ c ≡ 0 (mod n). □

III.12b. Case: char(k) | n and #J_{n} = n^{2}/p^{e}

Let p := char(k), and define e by p^{e} | n but p^{e+1} ∤ n. Since #μ_{n}(k̄) = n/p^{e}, the canonical lift H_{n} has order n^{3}/p^{2e}. If M, N ∈ H_{n} are such that M̄, N̄ generate J_{n}, and furthermore M̄ has order n while N̄ has order n/p^{e}, then each element of H_{n} has a unique expression in the form ζ_{n/p^{e}}^{a}M^{b}N^{c} with a ∈ Z/(n/p^{e})Z, b ∈ Z/nZ, and c ∈ Z/(n/p^{e})Z. We see easily that H_{n} is a central extension of J_{n} by μ_{n}(k̄).

III.12c. Case: char(k) | n and #J_{n} = n^{2}/p^{2e}

Set p and e as in §III.12b. The canonical lift H_{n} has order n^{3}/p^{3e}. If M, N ∈ H_{n} are such that M̄, N̄ generate J_{n}, then each element of H_{n} has a unique expression in the form ζ_{n/p^{e}}^{a}M^{b}N^{c} with a, b, c ∈ Z/(n/p^{e})Z. We again see easily that H_{n} is a central extension of J_{n} by μ_{n}(k̄).

III.13. Schrödinger-like representations of H_{n} when char(k) ∤ n

Assume char(k) ∤ n. We show in this section that there exist nice matrix representations over k̄ of both the canonical lift H_{n} ⊂ SL_{n}(k̄) and the non-canonical lifts H′_{n} ⊂ GL_{n}(k̄) of J_{n} ⊂ PGL_{n}(k̄), which were introduced in §III.12. We will see that the classical Schrödinger representation occurs for H_{n} when n is odd, while it occurs for one of the H′_{n} when n is even.

We first consider the canonical lift H_{n}. Fix choices M̄, N̄ ∈ J_{n} and M, N ∈ H_{n} as in the discussion in §III.12a. We saw in III.11.9 that the eigenvalues of M, N are each a coset of μ_{n}(k̄) inside k̄^{×}. By the definition of H_{n}, the product of the eigenvalues is 1. Therefore, when n is odd, both M and N have eigenvalues

{ 1, ζ_{n}, ζ_{n}^{2}, …, ζ_{n}^{n−1} },

while when n is even, both M and N have eigenvalues

{ ζ_{2n}, ζ_{2n}^{3}, ζ_{2n}^{5}, …, ζ_{2n}^{2n−1} }.

Over k̄, we can choose an eigenvector x for M corresponding to the first eigenvalue, and then use the basis { x, Nx, N^{2}x, …, N^{n−1}x } of eigenvectors for M (cf. proof of III.11.9) to obtain a new coordinate system for Pn−1 ; k̄. (Even though, when n is odd, x itself can be chosen to be defined over k, the remaining eigenvectors need not be defined over k.) For n odd, H_{n} will then have the classical Schrödinger representation

M = [\begin{matrix}1 \\ & ζ_{n} \\ & & ζ_{n}^{2} \\ & & & . & . & . \\ & & & & & & ζ_{n}^{n−1} \end{matrix}] , N = [\begin{matrix}& 1 \\ & & 1 \\ & & & . & . & . \\ & & & & & & 1 \\ 1 \end{matrix}] ,

while for n even we obtain

M = [\begin{matrix}ζ_{2n} \\ & ζ_{2n}^{3} \\ & & . & . & . \\ & & & & & ζ_{2n}^{2n−1} \end{matrix}] , N = [\begin{matrix}& 1 \\ & & . & . & . \\ & & & & & 1 \\ −1 \end{matrix}] .

We now consider one of the lifts H′_{n} ⊂ GL_{n}(k̄). For generators M′, N′ ∈ H′_{n}, if we define α, β ∈ μ_{n}(k̄) by (M′)^{n} = αM^{n} and (N′)^{n} = βN^{n} (cf. §III.12a), then in terms of the same basis used just now above, for n odd we obtain

M′ = [ \begin{matrix}^{n} & \sqrt{α} \\ & \sqrt[n]{α}ζ_{n} \\ & & . & . & . \\ & & & & & \sqrt[n]{α}ζ_{n}^{n−1} \end{matrix}] , N′ = [ \begin{matrix}^{n} & \sqrt{β} \\ & & . & . & . \\ ^{n} & \sqrt{β} \\ ^{n} & \sqrt{β} \end{matrix}] ,

while for n even we obtain

M′ = [ \begin{matrix}\sqrt[n]{α}ζ_{2n} \\ & \sqrt[n]{α}ζ_{2n}^{3} \\ & & . & . & . \\ & & & & & \sqrt[n]{α}ζ_{2n}^{2n−1} \end{matrix}] , N′ = [\begin{matrix}^{n} & \sqrt{β} \\ & & . & . & . \\ ^{n} & \sqrt{β} \\ − \sqrt[n]{β} \end{matrix}] .

We could instead scale the basis so that it is { x, N′x, …, (N′)^{n−1}x }. Then (N′)^{n}x is either βx (when n is odd) or −βx (when n is even), so for n odd we obtain

M′ = [ \begin{matrix}^{n} & \sqrt{α} \\ & \sqrt[n]{α}ζ_{n} \\ & & . & . & . \\ & & & & & \sqrt[n]{α}ζ_{n}^{n−1} \end{matrix}] , N′ = [\begin{matrix}& 1 \\ & & . & . & . \\ & & & & & 1 \\ β \end{matrix}] ,

while for n even we obtain

M′ = [ \begin{matrix}\sqrt[n]{α}ζ_{2n} \\ & \sqrt[n]{α}ζ_{2n}^{3} \\ & & . & . & . \\ & & & & & \sqrt[n]{α}ζ_{2n}^{2n−1} \end{matrix}] , N′ = [\begin{matrix}& 1 \\ & & . & . & . \\ & & & & & 1 \\ −β \end{matrix}] .

Thus, when n is even, we obtain the classical Schrödinger representation for the particular H′_{n} corresponding to (α, β) = (−1, −1).

Our use of a pair α, β ∈ μ_{n}(k̄) to “measure” how a lift H′_{n} deviates from H_{n} required us to then separately treat the cases n odd and n even in looking at Schrödinger-like representations. We can treat all values of n simultaneously by instead working with a pair α′, β′ ∈ μ_{n}(k̄) that measures how far we deviate from the classical Schrödinger representation.

Theorem III.13.1. Assume char(k) ∤ n. Let α′, β′ ∈ μ_{n}(k̄). Over k̄, there always exists a coordinate system so that the matrix group of order n^{3} generated by

[ \begin{matrix}^{n} & \sqrt{α′} \\ & \sqrt[n]{α′}ζ_{n} \\ & & . & . & . \\ & & & & & \sqrt[n]{α′}ζ_{n}^{n−1} \end{matrix}] and [\begin{matrix}& 1 \\ & & . & . & . \\ & & & & & 1 \\ β′ \end{matrix}]

represents one of the lifts H′_{n}. The classical Schrödinger representation occurs when (α′, β′) = (1, 1). For n odd, it has determinant 1 throughout and thus represents the canonical lift H_{n}; for n even, it has determinant ±1 throughout and thus does not represent H_{n}.

III.14. Concerning the Galois module structure of J_{C}[n] 43

III.14. Concerning the Galois module structure of J_{C}[n]

Theorem III.14.1. Assume char(k) ∤ n. If there exists a change-of-coordinates, defined over k, in which the canonical lift H_{n} has a Schrödinger-like representation (cf. §III.13), then there exists an isomorphism J_{C}[n] ≅ μ_{n} × Z/nZ.

Proof. The change-of-coordinates does not affect the abstract Galois module structure, so let H_{n} be the matrix group of order n^{3} generated by the two Schrödinger-like matrices M and N, as in §III.13. (Thus, if n is odd, the diagonal entries of M are 1, ζ_{2n}^{2}, ζ_{2n}^{4}, …, while if n is even, the diagonal entries of M are ζ_{2n}, ζ_{2n}^{3}, ζ_{2n}^{5}, . . ..) Each element of H_{n} has a unique expression in the form ζ_{n}^{a}M^{b}N^{c} with a, b, c ∈ Z/nZ, and each element of J_{C}[n](k̄) ≅ J_{n} has a unique expression in the form M̄^{b}N̄^{c}.

We can write down an isomorphism J_{n} [figure] μ_{n} × Z/nZ on k̄-valued points as follows: M̄^{b}N̄^{c} [figure] (ζ_{2n}^{2b}, c). We already know this to be an isomorphism of groups; as for Galois modules, we need worry only about the first coordinate. The result follows easily from the observation that ζ_{2n}^{2b} is always the same as the ratio of the first two diagonal entries of M^{b}. □

CHAPTER IV

An algorithm for the jacobian

Let k be a perfect field. As we saw in chapter III, every curve C of genus 1, defined over k, occurs as a non-degenerate degree n curve in P_{k}^{n−1} of genus 1. In this chapter, under the assumption char(k) ∤ n (regarding this, see §I.2e), we describe how to find equations for the jacobian J_{C} of C.

IV.1. An algorithm for the jacobian

Theorem IV.1.1. There exists an algorithm (described in this chapter) that takes as input a finite set of homogeneous equations; the algorithm checks whether the input defines a non-degenerate degree n curve in P_{k}^{n−1} of genus 1, where n−1 is the number of variables and n is coprime to char(k); if not, the algorithm terminates with an error; otherwise, it produces as output:

• coordinates of the n^{2} points of hyperosculation on C;

• equations describing the jacobian J_{C};

• coordinates of the k-rational origin on J_{C};

• polynomials describing the map j_{D}: C [figure] J_{C} of degree n^{2} that carries P [figure] [nP]−D, where D is the divisor class of hyperplane sections on C.

Proof. The rest of this chapter is the proof. □

Remarks. (1) To do all of this on a computer, we must of course have an encoding for elements of k and have algorithms for the field operations. In other words, k must be a computable field (cf. [BW93, §4.6]). In this chapter, we will not concern ourselves with such implementation-level details.

(2) More generally, we could take as input a finite set of homogeneous or inhomogeneous equations, not requiring them to define a non-degenerate degree n (smooth) curve in P_{k}^{n−1} or A_{k}^{n−1} of genus 1, and it is clear in principle that we would then need to do the following (but it is unclear how practical some of the steps are): we homogenize the equations and check that they define a variety of dimension is 1; then we blow up all the singularities to obtain a smooth curve of some degree n embedded in P_{k}^{m} for some m; we check that the genus is 1 and that n is coprime to char(k); then we intersect with a k-rational hyperplane to obtain a k-rational divisor D of degree n, then compute the map to projective space (cf. §II.4) determined by O(D). If n = 2, this map gives us a double cover of P_{k}^{1}, and we apply the formulas from [Wei54] (also in [AKM^{+}01]) to output everything stated in the theorem (the 4 points of hyperosculation correspond to the 4 points of ramification of the double cover, and these are easily determined). If n ≥ 3, the map to projective space gives us a new set of equations that meet the conditions of the theorem, and so we proceed with the algorithm described in this chapter.

IV.2. Vetting the input

The input to the algorithm is a finite set of homogeneous equations in a finite number of variables. There exist algorithms (see, e.g., [BS92]) for determining the dimension, genus, and degree of the scheme defined by the equations, as well as algorithms for determining whether a variety is smooth

IV.4. Finding H_{n} ⋊ { ±1 } 45

(by determining the dimension of the singular locus), and they are implemented in Macaulay 2 [GS] (cf. the code in table V.1).

If the dimension is ≠ 1, then the algorithm terminates with the message “input does not define a curve”.

If the genus is ≠ 1, then the algorithm terminates with the message “input curve does not have genus 1”.

If char(k) | n, where n is the degree, then the algorithm terminates with the message “degree of input curve is divisible by ground field characteristic”.

If the number of variables in the input equations is ≠ n − 1, then the algorithm terminates with the message “input curve is not of degree one higher than dimension of ambient projective space”.

If one of the input equations is linear, then the algorithm terminates with the message “input curve is degenerate (it lies in a hyperplane)”.

If the curve has a non-empty singular locus, then the algorithm terminates with the message “input curve is singular”.

IV.3. Describing all matrices that preserve the curve

Let I be the ideal generated by the input equations. Our curve is

C = Proj \frac{k[x_{0}, …, x_{n−1}]}{I},

but note that I may not be the ideal of the curve because I might not be saturated. (There exists algorithms for computing the saturation. We do not care whether I is saturated.) Let 𝔊 be a Gröbner basis for I with respect to some arbitrary (but fixed) monomial order on k[x_{0}, …, x_{n−1}].

Let A be an n × n matrix of variables a_{ij}. If x = [x_{0} : …: x_{n−1}] is a k̄-valued point of P_{k}^{n−1} that lies on C, then Ax may or may not lie on C. The condition for A to preserve C is that, for each f(x) ∈ I, where x is now a tuple of variables, also f(Ax) ∈ I. It is sufficient for this condition to hold on a set of generators for I. The ideal membership problem f(Ax) ∈ I is solved computationally using the Gröbner basis: we have f(Ax) ∈ I if and only if f(Ax) reduces to 0 upon division by 𝔊.

We proceed as follows: For each of the original generators f(x) of I given to us as part of the input, we divide f(Ax) by 𝔊 and, viewing the remainder as a polynomial in k(a_{ij})[x], set each remainder coefficient equal to 0. This process results in a system of equations in the a_{ij} whose solutions correspond to the matrices A that preserve C.

IV.4. Finding H_{n} ⋊ { ±1 }

To the system of equations in the a_{ij} obtained in the last step, add in the additional equation det(A) = 1. At this point, the equations define a 0-dimensional ideal. (The number of solutions will depend on the number of elliptic curve automorphisms the input curve admits over k̄. When the j-invariant of C is neither 0 nor 1728, then there will be 2n^{3} solutions.) The elements of H_{n} are then characterized as those solutions that act fixpoint-free on the curve (cf. §III.8). The various [−1] automorphisms (cf. §III.9) are characterized as those solutions that do have a fixpoint and have order 2 (modulo scalars).

IV.4a. Solving over a field extension

The 0-dimensional system defining the finitely many matrices must now be solved. There exist various techniques for this (cf. [CLO98, Ch. 2]). The simplest is elimination and extension. Slightly fancier is to obtain a Gröbner basis with respect to a favorite monomial order, and then use a Gröbner basis conversion procedure (cf. [CLO98, §2.3]) to land in the elimination/extension situation.

At any rate, a tedious but finite process will construct a normal field extension k(θ)/k and all the solution matrices, with all the entries expressed in terms of θ.

Once the solutions are in hand, we must identify H_{n}, and when n is even, we must also identify the [−1] automorphisms (we’ll need them to find the points of hyperosculation). Since fixpoints correspond to eigenvectors, we can use linear algebra to determine the fixpoints of each matrix, and the equations for C tell us whether any of the fixpoints lie on C.

For H_{n}, we hold on to the matrices that have no fixpoints, while for the [−1] automorphisms, we hold on to the matrices of order 2 (modulo scalars) and have a fixpoint. (In fact, by III.9.3, if we find one such matrix T, then they all have the form MT, where M ∈ H_{n}.)

Finally, by expanding all possible products of elements of H_{n}, we can find generators for H_{n}. In fact, since char(k) ∤ n, we will find two generators whose commutator is a primitive nth root of unity (cf. §III.12).

IV.4b. Practical improvements for finding H_{n}

Since char(k) ∤ n, the group H_{n} is characterized by the following properties: it admits a pair of matrices that act fixpoint-free on the curve and whose commutator is a primitive nth root of unity. Furthermore, each of the two generators admits n distinct fixpoints (not lying on the curve).

Thus, instead of solving for all matrices that preserve the curve, we need only solve for potentital generators of H_{n}. We can apply the following result to reduce the number of solutions, and then simply search for a pair of solutions whose commutator is a primitive nth root of unity, and which have no fixpoints on C.

Proposition IV.4.1. Assume char(k) ∤ n. If n is odd, and if M ∈ H_{n} is such that M̄ ∈ J_{n} has order n, then M^{n} = 1. If n is even, we instead have M^{n} = −1.

Proof. As we saw in III.11.9, the eigenvalues of M are { α, αζ_{n}, …, αζ_{n}^{n−1} }, for some α ∈ k̄^{×}. Thus, both the minimum and characteristic polynomials of M are ∏_{0≤i≤n−1}(X − αζ_{n}^{i}) = X + (−1)^{n}α^{n} ∏_{i} ζ_{n}^{i}. But the determinant is 1. Thus, for n odd, M satisfies X^{n} − 1, while for n even, M satisfies X^{n} + 1. □

Remark. In fact, when n is odd, H_{n} may be characterized as { A ∈ GL_{n}(k̄) : Ā ∈ J_{n} and A^{n} = 1 }. However, when n is even, we know of no such characterization.

Therefore, when n is odd, if one introduces the condition M^{n} = 1, one could leave off the det = 1 condition; however, it is unclear how—or even whether—the running time is affected by doing so (cf. V.2.1).

IV.4c. Practical improvements for finding the [−1] automorphisms

When n is even, we will need matrices corresponding to the [−1] automorphisms to find the points of hyperosculation on C.

When n is odd, we won’t need (as far as our algorithm for the jacobian is concerned) those matrices, but there may be independent interest in finding them.

We have already seen that the [−1] matrices are characterized by having a fixpoint on C and having order 2 in PGL_{n}(k̄), i.e., modulo scalars. By the determinant 1 condition, the only possible scalars are elements of μ_{n}(k̄).

Proposition. When n is odd, we can choose the [−1] matrices to satisfy A^{2} = 1.

Proof. Say we have found a matrix representing [−1] but A^{2} = ζ_{n}^{i}. If i is odd, then we replace A with ζ_{n}^{(n−i)/2}A. Then we will have A^{2} = 1 and det A = 1. If i is even, then we instead replace A with ζ_{n}^{(2n−i)/2}A. □

Proposition. When n is even, we can choose each [−1] matrix to satisfy either A^{2} = 1 or A^{2} = ζ_{n}.

Proof. Say A^{2} = ζ_{n}^{i}. If i is even, we replace A with ζ_{n}^{(n−i)/2}. If i is odd, we replace A with ζ_{n}^{(n+1−i)/2}. □

IV.5. Finding the points of hyperosculation

For this step, what we do depends on the parity of n.

IV.5a. When n is odd

The hyperplanes fixed by M will intersect our curve in the n^{2} points of hyperosculation. (The same is true for the hyperplanes fixed by N. See III.11.11.)

The hyperplanes fixed by M correspond to the eigenvectors of M^{t} (cf. §III.11). Thus, for example, if

M^{t}[α_{0}, …, α_{n−1}] = [α_{0}, …, α_{n−1}],

then

[α_{0}, …, α_{n−1}]^{t}M^{−1} = [α_{0}, …, α_{n−1}]^{t},

whence M fixes the hyperplane

α_{0}x_{0} + … + α_{n−1}x_{n−1} = 0.

Tossing the equation of a fixed hyperplane into our equations for I gives a 0-dimensional system whose solutions correspond to the points of intersection of the hyperplane with the curve. We solve this system by any convenient technique (cf. §IV.4a).

Repeating this process for each fixed hyperplane, we obtain the coordinates of the n^{2} points of hyperosculation. We print these out as part of the algorithm’s output.

IV.5b. When n is even

We find the points of hyperosculation by investigating the fixpoints of the [−1] automorphisms (cf. §III.9a). For each matrix T representing such an automorphism, either all of its fixpoints on C (there are four of them) are points of hyperosculation, or none of them are. For a given fixpoint, we look at all hyperplanes through that fixpoint, looking for the one with maximal intersection multiplicity with C at that fixpoint (this is the osculating hyperplane at that point). In fact, it is easy to see that the osculating hyperplane must itself be fixed by T, so we can restrict our search to the fixed hyperplanes of T.

If the osculating hyperplane meets C only at the fixpoint in question, then we have found a point of hyperosculation, and the other fixpoints of the matrix are also points of hyperosculation.

IV.6. Finding weighted equations for the jacobian

The action of H_{n} on the curve is a non-faithful representation of the action of J_{C}[n] on the curve. We have

J_{C} ≅ C/J_{C}[n] = C/H_{n}.

By basic principles of geometric invariant theory (cf. [Mum70, §II.7, §III.12] and [ABD^{+}64, §7]), we have

J_{C} ≅ Proj ((\frac{k(θ)[x_{0}, …, x_{n−1}]}{I k(θ)[x_{0}, …, x_{n−1}]})^{H_{n}})^{Gal(k(θ)/k)} .

Remark. In fact, since we only need to find J_{C} up to birational equivalence, we really only care that the Proj exhibited above has the correct function field. By the universal property of C/H_{n}, we can easily establish that its function field is

(K(C_{K})^{H_{n}})^{Gal(K/k)}, where K = k(θ).

We easily check that the Proj above has the same function field.

The following lemma helps us break the monster Proj expression into bite-size pieces. (In the lemma, we are interested in the cases G = H_{n} or G = Gal(k(θ)/k).)

Lemma. Let G be a finite group acting on a k-algebra R with trivial action on k, where k is a field and char(k) ∤ |G|. If I ⊆ R is an ideal with GI = I (so that G acts on R/I), then

(R/I)^{G} = R^{G}/I^{G}.

Proof. The averaging operator

∗: R [figure] R^{G}, r [figure] \frac{1}{|G|} ∑_{g∈G} r^{g}

is a homomorphism of R^{G}-modules. Taking G-invariants on the short exact sequence 0 [figure] I [figure] R [figure] R/I [figure] 0 gives the exact sequence

0 [figure] I^{G} [figure] R^{G} [figure] (R/I)^{G}.

If f + I ∈ (R/I)^{G}, then f − f^{g} ∈ I for all g ∈ G. Then \frac{1}{|G|}(∑_{g}(f − f^{g})) = f − f^{∗} also lies in I, whence the coset f + I is represented by f^{∗} ∈ R^{G}. Thus we have the short exact sequence of R^{G}-modules

0 [figure] I^{G} [figure] R^{G} [figure] (R/I)^{G} [figure] 0.
□

By the lemma, we have

J_{C} ≅ Proj \frac{(k(θ)[x_{0}, …, x_{n−1}]^{H_{n}})^{Gal(k(θ)/k)}}{((I k(θ)[x_{0}, …, x_{n−1}])^{H_{n}})^{Gal(k(θ)/k)}} .

(Observe the denominator may not be simply I. Certainly both H_{n} and Gal(k(θ)/k) preserve the curve, whence I is preserved as a whole, but the individual elements in I need not be invariant.)

By work of Hilbert and Noether, k(θ)[x_{0}, …, x_{n−1}]^{H_{n}} is finitely generated. There exist standard algorithms in the invariant theory of finite groups (see for example [Stu93]) for finding a system of generators.

Applying one of these algorithms, we obtain finitely many polynomials p_{i} in k(θ)[x_{0}, …, x_{n−1}] so that k(θ)[x_{0}, …, x_{n−1}]^{H_{n}} = k(θ)[p_{i}]. Since H_{n} itself is Gal(k(θ)/k)-invariant, it is in fact the case that the algorithms cited above will produce p_{i} that are themselves Gal(k(θ)/k)-invariant, and thus

(k(θ)[x_{0}, …, x_{n−1}]^{H_{n}})^{Gal(k(θ)/k)}

is simply k[p_{i}].

We end up with equations for J_{C} in terms of the generators p_{i} of the ring of invariants

(k(θ)[x_{0}, …, x_{n−1}]^{H_{n}})^{Gal(k(θ)/k)}. These actually express J_{C} as a closed subscheme of a weighted projective space, since the new variables p_{i} need not have degree 1.

One can work with closed subschemes of weighted projective spaces quite similarly to the way one works with them in usual projective spaces (see [Dol82]). However, weighted projective spaces do admit some pathologies (cf. [Dol82, §1.5])—for example, they can be singular. Following [Dol82, §1.1, Lemma], we formally eliminate all common factors from the degrees of the p_{i}. For example, if we have five generators of degrees (3; 6; 9; 9; 15), say, then we instead declare the degrees to be (1; 2; 3; 3; 5), but otherwise make no changes.

Remark. We make this simple formal degree change because much of the theory of weighted projective spaces assumes one has done so. One could also attempt to further clean up the description of the weighted space, such as by applying [Dol82, §1.3.1, Proposition].

The algorithm now outputs the equations and the (formally adjusted) degrees of the variables of the weighted projective space.

The map j_{D}: C [figure] J_{C} is simply the map P [figure] [p_{i}(P)]. The algorithm outputs the polynomials p_{i} to give an explicit description of j_{D}.

The k-rational origin on J_{C} is simply the image, under j_{D}, of any one of the points of hyperosculation on C. We determined those points earlier. Picking one, and evaluating the p_{i} there, the algorithm outputs the coordinates of the origin of J_{C}.

IV.7. Obtaining non-weighted equations for J_{C}

Using the affine covering described in [Har77, II.2.5b], we can find an affine patch of J_{C} containing the origin of J_{C}. Homogenizing the resulting equations leads to equations for J_{C} as a closed subscheme of a (non-weighted) projective space. The algorithm outputs these equations and the new coordinates for the origin.

Remark. An alternate approach is to take the function field of J_{C}, which can be read off from the weighted projective model, and express that field in the form k(α)[β]/⟨f(α, β)⟩, which leads to a non-weighted plane model of J_{C}.

Remark. If one’s goal is a Weierstrass model for J_{C}, one could now (or even prior to finding a non-weighted model) apply the Riemann–Roch algorithm (cf. §V.7c).

IV.8. Algorithm summary

Let k be a perfect field. The input to the algorithm is a finite set of homogeneous forms with coefficients in k.

(They are supposed to scheme-theoretically define a non-degenerate curve of degree n in P_{k}^{n−1} of genus 1, where n − 1 is the number of variables in the input, and char(k) ∤ n. But all of this is verified in the first step of the algorithm.)

(1) Determine the dimension, genus, and degree n of the scheme defined by the equations, as well as whether the singular locus is empty. Terminate with an error if the scheme does not have dimension 1, or if the curve does not have genus 1, or if n is a multiple of char(k), or if the number of variables differs from n − 1, or if the curve is singular, or if one of the equations is linear (whence the curve is degenerate).

(2) Let 𝔊 be a Gröbner basis (with respect to some fixed monomial order) for the ideal generated by the input equations. Let A be a generic n × n matrix. For each input equation f(x) = 0, divide f(Ax) by 𝔊 and set each coefficient of the remainder to 0, thus obtaining a system of equations in the a_{ij} that describe which matrices A preserve the input curve.

(3) With the additional equation det(A) = 1, solve for the finitely many solution matrices, expressing each matrix’s entries as elements in k(θ), where k(θ) is a finite normal field extension of k constructed during the solution process.

(4) Among the finitely many matrices, find the ones that act fixpoint-free on the curve, and find generators for that subgroup of matrices.

(5) If n is odd, determine the hyperplanes fixed by one of the generators, and intersect those hyperplanes with the curve to find the points of hyperosculation. If n is even, instead find the [−1]-matrices among the solutions found earlier, write down all their fixpoints, and then determine which of the fixpoints admit osculating hyperplanes that hyperosculate.

(6) Compute k-rational generators p_{i} for the subring k(θ)[x_{0}, …, x_{n−1}]^{H_{n}}, where H_{n} is the matrix group whose generators were found a couple steps ago. Compute k-rational generators for the ideal (I k(θ)[x_{0}, …, x_{n−1}])^{H_{n}}. These ideal generators serve as equations for J_{C} in a weighted projective space, whose coordinates are the p_{i}. Evaluating the p_{i} on the points of hyperosculation of C (see previous step) gives the coordinates of the origin on J_{C}.

(7) Grab an affine patch containing the origin of J_{C} and homogenize, thus obtaining equations for J_{C} in a (non-weighted) projective space.

CHAPTER V

Example: a Selmer cubic

Let k = Q, and let C ⊂ P_{Q}^{2} be the curve defined by F (x, y, z) = 0, where

F (x, y, z) = 3x^{3} + 4y^{3} + 5z^{3}.

That is,

C = Proj \frac{Q[x, y, z]}{⟨3x^{3} + 4y^{3} + 5z^{3}⟩}.  (V.1)

In this chapter, we will apply the algorithm from chapter IV to find the jacobian of (V.1), as well as related data (such as the points of hyperosculation on C). Afterwards, in §V.6, we will find some of the other data discussed in chapter III that was not necessary for finding J_{C}. Finally, in §V.7, we will generalize the results to the family

Proj \frac{k[x, y, z]}{⟨ax^{3} + by^{3} + cz^{3} + mxyz⟩},

where k is a perfect field with char(k) ≠ 3.

V.1. Vetting the input

It is easy to see that (V.1) is a smooth non-degenerate degree 3 curve in P_{Q}^{2} of genus 1: we check smoothness by verifying the matrix of partial derivatives of F has full rank everywhere on C; nondegeneracy follows because the generator F is not linear; the degree can be read off from F; the genus can be read off from the genus formula g = (d − 1)(d − 2)/2 for plane curves. The code in table V.1 will give us the same information.

          1  ringP2 = QQ[x,y,z]
          2  f=3*x^3+4*y^3+5*z^3
          3  idealC = ideal(f)
          4  ringC = ringP2 / idealC
          5  C = Proj ringC
          6  dim C
          7  codim singularLocus idealC
          8  degree C
          9  HH^1 OO_C

Table V.1. This Macaulay 2 code vets the input.

V.2. Finding H_{3}

We will now go about finding the 27 matrices that are the Q̄-valued points of the Heisenberg group H_{3} ⊂ SL_{3}, which, in its action on P_{Q}^{2}, preserves C and corresponds to the action of J_{C}[3] on C (cf. §III.12). It is characterized by three things: it preserves C, each matrix has determinant 1, and it acts fixpoint-free on C.

V.2. Finding H_{3} 51

First we will find conditions under which the generic matrix

A = [\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]

preserves C. In other words, whenever a point [x : y : z] lies on C, we require

Ā[x : y : z] = [a_{11}x + a_{12}y + a_{13}z : a_{21}x + a_{22}y + a_{23}z : a_{31}x + a_{32}y + a_{33}z]

to also lie on C, where Ā denotes the image of A in PGL_{3}(Q̄). That is, F (Ā[x : y : z]) must lie in the ideal ⟨F (x, y, z)⟩, where

F (Ā[x : y : z]) =

3(a_{11}x + a_{12}y + a_{13}z)^{3} + 4(a_{21}x + a_{22}y + a_{23}z)^{3} + 5(a_{31}x + a_{32}y + a_{33}z)^{3}. (∗)

This amounts to dividing (∗) by F and demanding the remainder to vanish, giving us 9 vanishing conditions on the coefficients of A:

F (Ā[x : y : z]) = (a_{11}^{3} + \frac{4}{3}a_{21}^{3} + \frac{5}{3}a_{31}^{3}) F (x, y, z) −

(4a_{11}^{3} − 3a_{12}^{3} + \frac{16}{3}a_{21}^{3} − 4a_{22}^{3} + \frac{20}{3}a_{31}^{3} − 5a_{32}^{3}) y^{3} + (9a_{12}^{2}a_{13} + 12a_{22}^{2}a_{23} + 15a_{32}^{2}a_{33}) y^{2}z +

(9a_{12}a_{13}^{2} + 12a_{22}a_{23}^{2} + 15a_{32}a_{33}^{2}) yz^{2} − (5a_{11}^{3} − 3a_{13}^{3} + \frac{20}{3}a_{21}^{3} − 4a_{23}^{3} + \frac{25}{3}a_{31}^{3} − 5a_{33}^{3}) z^{3} +

(9a_{11}^{2}a_{12} + 12a_{21}^{2}a_{22} + 15a_{31}^{2}a_{32}) x^{2}y + (9a_{11}^{2}a_{13} + 12a_{21}^{2}a_{23} + 15a_{31}^{2}a_{33}) x^{2}z +
(9a_{11}a_{12}^{2} + 12a_{21}a_{22}^{2} + 15a_{31}a_{32}^{2}) xy^{2} + (9a_{11}a_{13}^{2} + 12a_{21}a_{23}^{2} + 15a_{31}a_{33}^{2}) xz^{2} +
(18a_{11}a_{12}a_{13} + 24a_{21}a_{22}a_{23} + 30a_{31}a_{32}a_{33}) xyz.

         1  ring R = (0,a11,a12,a13,a21,a22,a23,a31,a32,a33),(x,y,z),dp;
         2  poly F = 3*x^3+4*y^3+5*z^3;
         3  ideal I = F;
         4  I = groebner(I);
         5  matrix A[3][3] = a11,a12,a13,a21,a22,a23,a31,a32,a33;
         6  matrix vars[3][1] = x,y,z;
         7  matrix newvars = A*vars;
         8  map substitution = R,newvars[1,1],newvars[2,1],newvars[3,1];
         9  poly substituted = substitution(F);
        10  poly reduced = reduce(substituted, I);
        11  matrix conditions = coef(reduced, x*y*z);
        12  conditions;

Table V.2. This Singular code determines the conditions for a matrix to preserve the Selmer cubic.

We next put these 9 conditions, along with the extra conditions det(A) = 1 and A^{3} = 1 (cf. IV.4.1) into an ideal in Q[a_{ij}] with elimination (lexicographic) ordering, and find a reduced Gröbner basis. For example, the code in table V.2 finds again the 9 conditions found previously by hand, and then the code in table V.3 produces the equations shown below.

Remark V.2.1. The A^{3} = 1 condition turns out to be crucial (this possibility is discussed in §IV.4b). Without that condition, the code would need to run an unknown amount of time— we gave up after waiting an hour. Even with A^{9} = 1 (a trivial condition—for both n even and n odd, we certainly have A^{n^{2}} = 1, since A^{n} is a scalar matrix), and with or without the det = 1 condition, we again gave up after an hour. But with A^{3} = 1 in place, with or without det = 1, the code runs almost instantaneously.

        13  ring S = 0,(a11,a12,a13,a21,a22,a23,a31,a32,a33),lp;
        14  ideal conditions_ideal;
        15
        16  int i; for (i = 1; i <= 9; i++) {
        17    setring R;
        18    poly condition = conditions[2,i];
        19    setring S;
        20    poly condition_in_S = imap(R,condition);
        21    condition_in_S = cleardenom(condition_in_S);
        22    conditions_ideal = conditions_ideal + condition_in_S;
        23    }
        24
        25  matrix AA = imap(R,A);
        26  poly determinant_one = 1 - det(AA);
        27  conditions_ideal = conditions_ideal + determinant_one;
        28
        29  LIB "matrix.lib";  // Lets us call power(), unitmat(), ...
        30  matrix AA_cubed = power(AA,3) - unitmat(3);
        31  int j; for (i = 1; i <= 3; i++) {
        32    for (j = 1; j <= 3; j++) {
        33      conditions_ideal = conditions_ideal + AA_cubed[i,j];
        34      }
        35    }
        36
        37  option(redSB); // so that ``groebner'' returns reduced result...
        38  conditions_ideal = groebner(conditions_ideal);
        39  conditions_ideal;

Table V.3. This Singular code (continuation from previous table) determines the conditions, in elimination order, for a matrix to satisfy three conditions: preserve the Selmer cubic, have determinant 1, and have its cube be 1.

125a_{32}^{10} − 64a_{32} = 0,  a_{33}^{4} − a_{33} = 0, a_{32}a_{33} = 0, a_{31}a_{33} = 0,
8000a_{31}^{9} + 3375a_{32}^{9} + 1728a_{33}^{3} − 1728 = 0, 12a_{23}^{3} − 25a_{31}^{3} = 0, a_{31}a_{32} = 0, a_{23}a_{33} = 0,
125a_{21}a_{32}^{9} − 64a_{21} = 0,  a_{22}a_{33}^{3} − a_{22} = 0, a_{23}a_{32} = 0, a_{22}a_{32} = 0,
12a_{13} − 25a_{21}^{2}a_{32}^{5} = 0,  a_{22}^{3} − a_{33}^{3} = 0, a_{22}a_{31} = 0, a_{22}a_{23} = 0,
9a_{12} − 20a_{23}^{2}a_{31}^{5} = 0, 16a_{21}^{3} − 15a_{32}^{3} = 0, a_{21}a_{33} = 0, a_{21}a_{31} = 0,
a_{11} − a_{22}^{2}a_{33}^{2} = 0, a_{21}a_{23} = 0, a_{21}a_{22} = 0.

These equations define a 0-dimensional ideal in Q[a_{ij}]. There exist computer algorithms, such as the ones in Singular’s library zeroset lib, that will construct a field extension containing all solutions and then list the solutions. Waiting on the computer to do this exhausted our patience, so we simply solved it by hand.

Let K = Q(ζ_{3}, \sqrt[3]{3}, \sqrt[3]{4}, \sqrt[3]{5}), where ζ_{3} is a primitive cube root of unity, i.e., ζ_{3}^{2} + ζ_{3} + 1 = 0, and each of the other symbols satisfies the obvious equation, e.g., (\sqrt[3]{4})^{3} = 4. A symbol such as \sqrt[3]{−4/5} is shorthand for −\sqrt[3]{4}/\sqrt[3]{5}.

When we go about finding all matrices in the solution set (the details are left to the reader), we discover that there are 9 solutions (modulo cube roots of unity): 1, M, M^{2}, N, MN, M^{2}N, N^{2}, MN^{2}, M^{2}N^{2}, where

M = [\begin{matrix}1 & 0 & 0 \\ 0 & ζ_{3} & 0 \\ 0 & 0 & ζ_{3}^{2} \end{matrix}] and N = [\begin{matrix}0 & \sqrt[3]{4/3} & 0 \\ 0 & 0 & \sqrt[3]{5/4} \\ \sqrt[3]{3/5} & 0 & 0 \end{matrix}] .

But we didn’t really have to find them all. We merely need to find two generators for the Heisenberg group H_{3}. Thus we must find a pair of matrices that act fixpoint-free on C and whose commutator is a primitive cube root of unity.

V.4. The curve underlying the jacobian J_{C} 53

It is easy to check that the two matrices M and N satisfy the commutator condition. The fixpoints of M and N in P_{Q}^{2}(Q̄) correspond to the eigenspaces of M and N (cf. §III.11). Using standard linear algebra techniques for finding eigenvalues and eigenvectors, we determine that M fixes

[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1],

while N fixes

[1 : \sqrt[3]{3/4} : \sqrt[3]{3/5}],
[1 : ζ_{3} \sqrt[3]{3/4} : ζ_{3}^{2} \sqrt[3]{3/5}],
[1 : ζ_{3}^{2} \sqrt[3]{3/4} : ζ_{3} \sqrt[3]{3/5}].

Substituting these points into the equation defining C verifies that the action of M and N on C is indeed fixpoint-free.

V.3. Points of hyperosculation and the hyperplane configuration

The hyperplanes fixed by M and N correspond to the eigenvectors of M^{t} and N^{t} (cf. §III.11). We determine that M fixes the hyperplanes

x = 0, y = 0, z = 0,

while N fixes the hyperplanes

x + \sqrt[3]{4/3}y + \sqrt[3]{5/3}z = 0,
x + ζ_{3}^{2} \sqrt[3]{4/3}y + ζ_{3} \sqrt[3]{5/3}z = 0,
x + ζ_{3} \sqrt[3]{4/3}y + ζ_{3}^{2} \sqrt[3]{5/3}z = 0.

Intersecting either set of hyperplanes with C produces the 9 points of hyperosculation (cf. §III.11.8):

[0 : 1 : \sqrt[3]{−4/5}], [0 : 1 : ζ_{3} \sqrt[3]{−4/5}], [0 : 1 : ζ_{3}^{2} \sqrt[3]{−4/5}],
[\sqrt[3]{−5/3} : 0 : 1], [ζ_{3} \sqrt[3]{−5/3} : 0 : 1], [ζ_{3}^{2} \sqrt[3]{−5/3} : 0 : 1],
[1 : \sqrt[3]{−3/4} : 0], [1 : ζ_{3} \sqrt[3]{−3/4} : 0], [1 : ζ_{3}^{2} \sqrt[3]{−3/4} : 0].

The rows correspond to the hyperplanes fixed by M, while the columns correspond to the hyperplanes fixed by N. We see here the arrangement and intersection pairing described in §III.11.11.

Remark. Classically, the points of hyperosculation on a plane cubic are found using the Hessian. We say more about this in §V.6c.

V.4. The curve underlying the jacobian J_{C}

The matrices M and N generate an abelian group J_{n} ⊂ PGL_{3}(K) whose elements correspond to J_{C}[3](Q̄). Thus J_{C}[3] is defined (element-wise) over K.

The same matrices generate a group of order 27 in SL_{3}(K). Let H_{3} ⊂ SL_{3} be the subgroup with those K -valued points. By the material in §IV.6, we have

J_{C} ≅ Proj \frac{Q[x, y, z] ∩ K[x, y, z]^{H_{3}(K)}}{I ∩ K[x, y, z]^{H_{3}(K)}} .

To determine the latter more explicitly, we start by looking for an explicit description of K[x, y, z]^{H_{3}(K)}. We know from general principles (see [Stu93]) that there will be three algebraically

54 V. Example: a Selmer cubic

 1  (*MonomialList generates all monomial exponent vectors
 2    satisfying the given degree constraints.*)
 3  MonomialList[varcount_, minweight_, maxweight_] :=
 4      Module[
 5        {v, i, r, keepgoing},
 6        v = Table[maxweight, {varcount}];
 7        r = {};
 8        keepgoing = True;
 9        While[keepgoing,
10          If[minweight <= (Plus @@ v) && (Plus @@ v) <= maxweight,
11            r = Append[r, v];
12            ];
13          (* Subtract 1 from first index in varlist v. *)
14          (* If result is negative, replace with maxweight,
15             and move right. If we run out of variables, quit. *)
16          i = 1;
17          While[--v[[i]] < 0,
18            v[[i]] = maxweight;
19            ++i;
20            If[i > varcount,
21              keepgoing = False; Break[];];
22            ];
23          ];
24        r
25        ];
26
27  (*Reynolds averages the value of the function on the matrixgroup orbit of varlist.*)
28  ReynoldsOperator[matrixgroup_List, f_, varlist_List] :=
29      Apply[Plus, Map[Apply[f, #.varlist] &, matrixgroup]]/Length[matrixgroup];
30
31  (*GeneratingInvariants applies Reynolds to all monomials of a given degree.*)
32  GeneratingInvariants[matrixgroup_List, varlist_List, degree_Integer] :=
33      Module[
34        {m = MonomialList[Length[varlist], degree, degree]},
35        Table[
36          ReynoldsOperator[matrixgroup,
37            Function[varlist, Times @@ Apply[Power, {varlist, m[[i]]}]],
38            varlist],
39          {i, Length[m]}]
40        ];
41
42  InvertedCharPoly[matrix_List, var_] :=
43      Det[IdentityMatrix[Length[matrix]] - var*matrix];
44  MolienSeries[matrixgroup_List, var_] :=
45    (1/Length[matrixgroup])*Plus @@ Map[1/InvertedCharPoly[#, var] &, matrixgroup]]
46
47  HeisenbergGroup[m_List, n_List] :=
48    Module[{d = Dimensions[m][[1]], z = m.n.Inverse[m].Inverse[n]},
49      Flatten[Table[
50          MatrixPower[z, i].MatrixPower[m, j].MatrixPower[n, k], {i, d}, {j,
51            d}, {k, d}], 2]]

Table V.4. This Mathematica code defines functions we’ll need in the next listing.

independent invariants (known as primary invariants), and possibly some additional dependent invariants (secondary invariants). The Molien series,

Φ_{H_{3}}(t) = \frac{1}{27} ∑_{σ∈H_{3}(K)} \frac{1}{det(id − tσ)}

= \frac{1}{27}(\frac{1}{(1 − t)^{3}} + \frac{1}{(1 − ζ_{3}t)^{3}} + \frac{1}{(1 − ζ_{3}^{2}t)^{3}} + \frac{24}{1 − t^{3}} ) = \frac{1 − t^{3} + t^{6}}{(1 − t^{3})^{3}}

= 1 + 2t^{3} + 4t^{6} + 7t^{9} + 11t^{12} + 16t^{15} + 22t^{18} + 29t^{21} + 37t^{24} + ⋯,

tells us that there are two linearly independent invariants of degree 3, four of degree 6, seven of degree 9, and so on.

(The Mathematica code in table V.5, relying on the functions from table V.4, will compute the Molien series above as well as what we are about to calculate.)

The three possible products of the two invariants of degree 3 do not account for all four linearly independent invariants of degree 6. Hence we start by looking in degree 3 and degree 6. Applying

        52  cuberoot1 = Root[(#^2 + # + 1), 1];
        53  cuberoot3 = Root[(#^3 - 3), 1];
        54  cuberoot4 = Root[(#^3 - 4), 1];
        55  cuberoot5 = Root[(#^3 - 5), 1];
        56  vars = {x,y,z};
        57  MM = DiagonalMatrix[Map[(cuberoot1^#) &, Range[3]]];
        58  NN = RotateLeft[
        59        DiagonalMatrix[{cuberoot3/cuberoot5, cuberoot4/cuberoot3, cuberoot5/cuberoot4}]];
        60  MM // Simplify // MatrixForm
        61  NN // Simplify // MatrixForm
        62
        63  h = HeisenbergGroup[MM, NN];
        64  ms = MolienSeries[h, t]
        65  ms = Simplify[ms]
        66  Series[ms, {t, 0, 30}]
        67
        68  Factor[Union[Simplify[GeneratingInvariants[h, vars, 3]]]]
        69  Factor[Union[Simplify[GeneratingInvariants[h, vars, 6]]]]
        70
        71  P1 = x y z;
        72  P2 = 3x^3 + 4y^3 + 5z^3;
        73  P3 = 9x^6 + 16y^6 + 25z^6;
        74  GroebnerBasis[{P1-y1,P2-y2,P3-y3}, {x,y,z,y1,y2,y3}, {x,y,z}]
        75
        76  Simplify[ms*((1 - t^3)(1 - t^3)(1 - t^6))]
        77  GeneratingInvariants[h, vars, 9] // Simplify // Factor // Simplify // Union // Factor
        78
        79  P4 = 27x^9 + 64y^9 + 125z^9;
        80  GroebnerBasis[{P1-y1,P2-y2,P3-y3,P4-y4}, {x,y,z,y1,y2,y3,y4}, {x,y,z}]
        81
        82  P4 = 48x^3y^6 + 45x^6z^3 + 100y^3z^6;
        83  syzygy = GroebnerBasis[{P1-y1,P2-y2,P3-y3,P4-y4}, {x,y,z,y1,y2,y3,y4}, {x,y,z}]
        84
        85  gg = GroebnerBasis[{P1-y1,P2-y2,P3-y3,P4-y4}, {x,y,z,y1,y2,y3,y4}];
        86  PolynomialReduce[3x^3 + 4y^3 + 5z^3, gg, {x,y,z,y1,y2,y3,y4}]
        87
        88  syzygy /. {y2 -> 0}

Table V.5. This Mathematica code (relying on functions from previous table) calculates the Molien series, the primary and secondary invariants, and the final syzygy that gives the model for J_{C}.

the Reynolds operator to all degree 3 and all degree 6 monomials gives the following list of invariants:

xyz,  3x^{3} + 4y^{3} + 5z^{3},  xyz(3x^{3} + 4y^{3} + 5z^{3}),
x^{2}y^{2}z^{2},  9x^{6} + 16y^{6} + 25z^{6},  12x^{3}y^{3} + 15x^{3}z^{3} + 20y^{3}z^{3}.

Two invariants of degree 3 together with one invariant of degree 6 must compose a set of algebraically independent invariants. We arbitrarily choose

\begin{aligned}P_{1} & := xyz \\ P_{2} & := 3x^{3} + 4y^{3} + 5z^{3} \\ P_{3} & := 9x^{6} + 16y^{6} + 25z^{6} \end{aligned}

We verify their algebraic independence by introducing slack variables s_{1}, s_{2}, s_{3} with an elimination ordering on monomials and compute a Gröbner basis for the ideal ⟨P_{1} − s_{1}, P_{2} − s_{2}, P_{3} − s_{3}⟩ in Q[x, y, z, s_{1}, s_{2}, s_{3}] and then intersect with Q[s_{1}, s_{2}, s_{3}] to read off any syzygies: there are none. In other words, the ring homomorphism from Q[s_{1}, s_{2}, s_{3}] to Q[P_{1}, P_{2}, P_{3}] given by s_{i} [figure] P_{i} has trivial kernel. (All of this is explained quite nicely in [CLO97].

We can furthermore easily see that (0, 0, 0) is the only solution to P_{1} = 0, P_{2} = 0, P_{3} = 0; whence, by the Nullstellensatz, the radical of ⟨P_{1}, P_{2}, P_{3}⟩ is ⟨x, y, z⟩. Thus K[x, y, z] is integral over K[P_{1}, P_{2}, P_{3}], whence also K[x, y, z]^{H_{3}(K)} is integral over K[P_{1}, P_{2}, P_{3}], and therefore is finitely generated as a module over K[P_{1}, P_{2}, P_{3}]. (In fact, as explained in [Stu93], K[x, y, z]^{H_{3}(K)} will be free over K[P_{1}, P_{2}, P_{3}], since the subring of invariants is Cohen–Macaulay.)

Thus P_{1}, P_{2}, P_{3} shall serve as primary invariants. They generate a graded subring of the subring of invariants, and their Hilbert series is

\frac{1}{1 − t^{3}} \frac{1}{1 − t^{3}} \frac{1}{1 − t^{6}} ,

which forms the denominator of the Hironaka form of our earlier Molien series:

Φ_{H_{3}}(t) = \frac{1 + t^{9}}{(1 − t^{3})(1 − t^{3})(1 − t^{6})}.

Thus there is one secondary invariant, in degree 9, which we now locate. Applying the Reynolds operator to all degree 9 monomials gives us:

x^{2}y^{2}z^{2}(3x^{3}+4y^{3}+5z^{3}), 36x^{6}y^{3}+80y^{6}z^{3}+75x^{3}z^{6},
xyz(9x^{6}+16y^{6}+25z^{6}), 48x^{3}y^{6}+45x^{6}z^{3}+100y^{3}z^{6},
x^{3}y^{3}z^{3},  27x^{9}+64y^{9}+125z^{9},  xyz(12x^{3}y^{3}+15x^{3}z^{3}+20y^{3}z^{3}).

If we compute a Gröbner basis for the ideal

⟨P_{1} − s_{1}, P_{2} − s_{2}, P_{3} − s_{3}, 27x^{9} + 64y^{9} + 125z^{9} − s_{4}⟩

and intersect with Q[s_{1}, s_{2}, s_{3}, s_{4}], we obtain the syzygy

−360s_{1}^{3} + s_{2}^{3} − 3s_{2}s_{3} + 2s_{4};

since this syzygy is linear in s_{4}, it tells us that 27x^{9} + 64y^{9} + 125z^{9} is linearly dependent on other degree 9 invariants expressible in terms of our previous choices. We next try

P_{4} := 48x^{3}y^{6} + 45x^{6}z^{3} + 100y^{3}z^{6},

and this time the syzygy is not linear in s_{4}:

{259200s_{1}^{6} − 960s_{1}^{3}s_{2}^{3} + s_{2}^{6} + 1440s_{1}^{3}s_{2}s_{3} − 3s_{2}^{4}s_{3} \atop + 3s_{2}^{2}s_{3}^{2} − s_{3}^{3} + 1440s_{1}^{3}s_{4} − 4s_{2}^{3}s_{4} + 4s_{2}s_{3}s_{4} + 8s_{4}^{2}}.

In other words,

K[x, y, z]^{H_{3}(K)} ≅ \frac{K[s_{1}, s_{2}, s_{3}, s_{4}]}{⟨259200s_{1}^{6}−960s_{1}^{3}s_{2}^{3}+s_{2}^{6}+1440s_{1}^{3}s_{2}s_{3}−3s_{2}^{4}s_{3} ; +3s_{2}^{2}s_{3}^{2}−s_{3}^{3}+1440s_{1}^{3}s_{4}−4s_{2}^{3}s_{4}+4s_{2}s_{3}s_{4}+8s_{4}^{2}⟩},

where this is an isomorphism of graded rings so long as we keep in mind a weighted grading on the right: the variables s_{1}, s_{2}, s_{3}, s_{4} have degrees 3, 3, 6, 9 (corresponding to our choices of P_{1}, P_{2}, P_{3}, P_{4}).

Remark. Although irrelevant to our calculations, it is helpful to understand what the primary and secondary invariants have to do with the structure of the ring of invariants. As already mentioned, the ring K[x, y, z]^{H_{3}(K)} is Cohen–Macaulay, and thus it is a finitely generated free module, generated by the secondary invariants over the subring generated by the algebraically independent primary invariants:

K[x, y, z]^{H_{3}(K)} = K[P_{1}, P_{2}, P_{3}] ⊕ P_{4}K[P_{1}, P_{2}, P_{3}].

Having determined K[x, y, z]^{H_{3}(K)}, we return to the problem of finding J_{C}. Observe that our choices are such that the equation of the curve is given by P_{2} = 0. Furthermore, since H_{3}(K) is defined over Q, the Molien series and the invariants calculated above were each Gal(K/Q)-invariant. Thus we have

Q[x, y, z] ∩ K[x, y, z]^{H_{3}(K)} ≅ \frac{Q[s_{1}, s_{2}, s_{3}, s_{4}]}{⟨259200s_{1}^{6}−960s_{1}^{3}s_{2}^{3}+s_{2}^{6}+1440s_{1}^{3}s_{2}s_{3}−3s_{2}^{4}s_{3} ; +3s_{2}^{2}s_{3}^{2}−s_{3}^{3}+1440s_{1}^{3}s_{4}−4s_{2}^{3}s_{4}+4s_{2}s_{3}s_{4}+8s_{4}^{2}⟩},

and substituting s_{2} = 0 corresponds to working modulo I, so that we have established

J_{C} ≅ Proj \frac{Q[s_{1}, s_{3}, s_{4}]}{⟨259200s_{1}^{6} − s_{3}^{3} + 1440s_{1}^{3}s_{4} + 8s_{4}^{2}⟩}.

We have expressed the curve underlying J_{C} as a curve of degree 18 in a weighted projective plane P_{Q}^{(3;6;9)}. By the general theory of weighted projective planes (see the end of §IV.6), we formally

relabel the weights to be (1; 2; 3), so that J_{C} is now a curve of degree 6 in P_{Q}^{(1;2;3)}.

The Q-rational group law origin on J_{C} corresponds to the points of hyperosculation on C. Since the functions s_{1}, s_{3}, s_{4} are invariant on those n^{2} points, we can obtain coordinates for the origin on J_{C} by picking any one of the points of hyperosculation and applying the map (cf. II.2.4)

j_{D}: [x : y : z] [figure][P_{1}(x, y, z) :: P_{3}(x, y, z) :: P_{4}(x, y, z)],

where we use the notation [· :: · :: ·] on weighted homogeneous coordinates to remind us of the unusual equivalence relation: [x :: y :: z] ∼ [λx :: λ^{2}y :: λ^{3}z].

For example, one of the points of hyperosculation is [0 : 1 : \sqrt[3]{−4/5}]. Evaluating j_{D} at this point gives us the group law origin on J_{C}:

O_{J_{C}} = [0 :: 2 :: 1].

Remark. In fact, evaluating [P_{1} :: P_{3} :: P_{4}] at each of the 9 points of hyperosculation leads to [0 :: 32 :: 64], [0 :: 50 :: 125], and [0 :: 18 :: 27]. But each of those is just [0 :: 2 :: 1].

In summary, we have determined the jacobian of (V.1) to be

J_{C} ≅ Proj \frac{Q[r, s, t]}{⟨259200r^{6} − s^{3} + 1440r^{3}t + 8t^{2}⟩},  (V.2)

where variables r, s, t have degrees (1; 2; 3), and the origin of the group law is the Q-rational point [0 :: 2 :: 1]. We obtained J_{C} as an elliptic curve of degree 6 in the weighted projective plane P_{Q}^{(1;2;3)} = Proj(Q[r, s, t]).

V.5. Obtaining a non-weighted model for J_{C}

We will now obtain a non-weighted version of (V.2) using the approach from §IV.7. The weighted projective plane P_{Q}^{(1;2;3)} is covered by the following three affine patches (see [Har77, II.2.5b])

Spec Q[\frac{s}{r^{2}} , \frac{t}{r^{3}} ] ≅ Spec Q[u, v] (open subset where r ≠ 0),
Spec Q[\frac{r^{2}}{s} , \frac{rt}{s^{2}} , \frac{t^{2}}{s^{3}} ] ≅ Spec \frac{Q[u, v, w]}{⟨uw − v^{2}⟩} (open subset where s ≠ 0),
Spec Q[\frac{r^{3}}{t} , \frac{rs}{t}, \frac{s^{3}}{t^{2}} ] ≅ Spec \frac{Q[u, v, w]}{⟨uw − v^{3}⟩} (open subset where t ≠ 0),

where u, v, w are indeterminates. Thus we have three different ways of obtaining an affine model for J_{C}, but only two of those models (the latter two above) contain the image of the Q-rational origin [0 :: 2 :: 1] on J_{C}.

Let us use the affine patch corresponding to s ≠ 0. Dividing the defining expression in (V.2) through by s^{3} leads to

Spec \frac{Q[u, v, w]}{⟨uw − v^{2}, 259200u^{3} − 1 + 1440uv + 8w⟩} ⊂ A_{Q}^{3},

where the Q-rational origin is now (0, 0, 1/8). We can of course homogenize this if we prefer a projective model for J_{C}.

We have finished applying the algorithm from chapter IV to F (x, y, z) = 3x^{3} + 4y^{3} + 5z^{3} = 0. In the next section, we will say some additional things about that example. Then, in §V.7, we will tackle the family ax^{3} + by^{3} + cz^{3} + mxyz = 0.

V.6. Musings

V.6a. Finding the 2-torsion on (C, O)

We will determine the matrix T corresponding to [−1] on the elliptic curve (C, O), where O is the following point of hyperosculation:

O = [0 : 1 : \sqrt[3]{−4/5}].

Thus we seek a matrix A of determinant 1 that preserves C and also fixes O, and furthermore its square should be a scalar matrix (so that the matrix has order 2 on the curve). But if, say, A^{2} = ζ_{3}, then (ζ_{3}A)^{2} = 1, and det(ζ_{3}A) = 1. Thus we can find an A that satisfies A^{2} = 1. To do this, we repeat the calculations from tables V.2 and V.3, only this time we replace the condition A^{3} = 1 with A^{2} = 1. To preserve O, we discover we must have, among other things, the additional condition 5a_{12}^{3} = 4a_{13}^{3}, which already simplifies matters sufficiently to determine

T = [\begin{matrix}−1 & 0 & 0 \\ 0 & 0 & \sqrt[3]{−5/4} \\ 0 & \sqrt[3]{−4/5} & 0 \end{matrix}] .

The three eigenvalues of T are 1, −1, −1. There is one isolated fixpoint in P_{Q}^{2}, and one line’s worth of fixpoints in P_{Q}^{2}, which gives us 3 fixpoints on C. Since T must have 4 fixpoints on C, the isolated fixpoint must lie on C. Indeed, by computing eigenvectors, we determine that the fixpoint corresponding to eigenvalue +1 is O, while the fixed line is spanned by [1 : 0 : 0] and [0 : 1 : \sqrt[3]{4/5}]. That line cuts C in the following three points:

[\sqrt[3]{−8/3} : 1 : \sqrt[3]{4/5}], [ζ_{3} \sqrt[3]{−8/3} : 1 : \sqrt[3]{4/5}], [ζ_{3}^{2} \sqrt[3]{−8/3} : 1 : \sqrt[3]{4/5}].

These must be the three points of non-trivial 2-torsion on (C, O). Note that they are not points of hyperosculation (cf. §III.9).

V.6b. Obtaining a Weierstrass model for J_{C}

As mentioned in §I.1b, once we have a curve of genus 1 together with a rational point (i.e., an elliptic curve), we can always apply the Riemann–Roch algorithm to obtain a Weierstrass model for that curve (cf. §V.7c). In the present example, it turns out there is an easier way.

At the start of §V.5, we chose an affine patch containing the Q-rational origin on J_{C}. If we instead work with an affine patch that does not contain that point, then we will obtain an affine model with the origin “at infinity”.

Dividing the defining expression in (V.2) through by r^{6} leads to the affine model

Spec \frac{Q[u, v]}{⟨259200 − u^{3} + 1440v + 8v^{2}⟩} ⊂ A_{Q}^{2}.

We knew this would give us an affine plane model, but it so happens we already ended up with a Weierstrass model! If we now apply the substitution u [figure] 2X, v [figure] \frac{1}{2}(Y − 180), and clear off the common factor 2 from the result, we end up with

Y^{2} = 4X^{3} − 97200.  (V.3)

In the cases n = 2, 3, 4, formulas for a Weierstrass model of J_{C} appear in [AKM^{+}01]. (For n = 5, see [Fis].) If we apply those formulas to (V.1), we also end up with (V.3) above.

V.6c. The classical Hessian

In §V.3, we found the 9 points of hyperosculation on our plane cubic (V.1) by finding the hyperplane sections fixed by a generator for H_{3}.

Classically (cf. [Sil99, Ex. III.3.9]), the 9 points of hyperosculation (called flex points) on a plane cubic F (x, y, z) = 0 are obtained by intersecting the cubic with its Hessian H(x, y, z) = 0, where

H(x, y, z) := |\begin{matrix}\frac{∂^{2}F}{∂x^{2}} & \frac{∂^{2}F}{∂x ∂y} & \frac{∂^{2}F}{∂x ∂z} \\ \frac{∂^{2}F}{∂y ∂x} & \frac{∂^{2}F}{∂y^{2}} & \frac{∂^{2}F}{∂y ∂z} \\ \frac{∂^{2}F}{∂z ∂x} & \frac{∂^{2}F}{∂z ∂y} & \frac{∂^{2}F}{∂z^{2}} \end{matrix} | .

V.7. Tackling the family ax^{3} + by^{3} + cz^{3} + mxyz = 0 59

Even without knowing any of the general theory of the Hessian, we can see quite easily that the Hessian in our current example behaves as the theory says it must. For our F, the Hessian H is just (a scalar multiple of) of the invariant P_{1} from §V.4. In calculating the origin on J_{C}, we saw that P_{1} vanishes at the 9 points of hyperosculation; on the other hand, by Bézout’s theorem (see [Har77, I.7.8]), it cannot vanish elsewhere on C.

In fact, we saw in §V.4 that the space of cubic invariants is 2-dimensional, spanned by F and P_{1}. Therefore, other than scalar multiples of F, any cubic invariant will intersect C in the points of hyperosculation. A more enlightening way of seeing this goes as follows.

Invariants of any degree that do not vanish identically on C will necessarily intersect C in a union of 3-torsion packets. If the invariant is defined over Q, then its intersection with C is also defined over Q. In particular, by Bézout, a cubic invariant will intersect C in a single 3-torsion packet defined over Q; therefore, if we can show that the hyperosculation packet is the unique such packet, then it follows that any cubic invariant must intersect C in the hyperosculation packet.

A change-of-variables to (V.3) yields Y^{2} = X^{3} − 24300. Feeding [0,0,0,0,-24300] to mwrank tells us that J_{C} has rank 0, while feeding

                      e=ellinit([0,0,0,0,-24300]);elltors(e)

to GP/Pari tells us that J_{C} has a single point of torsion. In short, the hyperosculation packet is the unique 3-torsion packet on C that is defined over Q.

V.7. Tackling the family ax^{3} + by^{3} + cz^{3} + mxyz = 0

Let k be a perfect field with char(k) ≠ 3 (concerning this restriction, cf. §I.2e). Set

C = Proj \frac{k[x, y, z]}{⟨ax^{3} + by^{3} + cz^{3} + mxyz⟩},  (V.4)

where a, b, c, m ∈ k. The condition for C to be smooth is abc(27abc + m^{3}) ≠ 0.

V.7a. History

The family (V.4) has been studied extensively (often without the mxyz term). It was known classically that a k-rational point on (V.4) leads to a non-trivial k-rational point on the elliptic curve

E = Proj \frac{k[X, Y, Z]}{⟨X^{3} + Y^{3} + abcZ^{3} + mXY Z⟩}, with origin O_{E} = [1 : −1 : 0].  (V.5)

Formulas for this are attributed to Sylvester in [Cal92, §3] and to Euler in [Sel51, §I.2]. See also [Cas91, §18, Lemma 1].

That (V.5) is the jacobian of (V.4) has probably been known for a long time—the earliest reference I could find, for m = 0 and k = Q(ζ_{3}), is [Cas91, §20, Ex. 3]. The proof outlined there works in general, and goes as follows. Over k(ζ_{3}, \sqrt[3]{a}, \sqrt[3]{b}) we can write down the isomorphism ϕ: (V.4) [figure](V.5) given by X = \sqrt[3]{a} x, Y = \sqrt[3]{b} y, and Z = (\sqrt[3]{ab})^{−1}z. Then ξ_{σ} := ϕ^{σ} ∘ ϕ^{−1} is an element of H^{1}(G_{k}, E(k̄) ⋊ Aut(E, O_{E})). It is easy to verify that each ξ_{σ} is a fixpoint-free automorphism of E, whence ξ ∈ H^{1}(G_{k}, E(k̄)). Therefore, C is a principal homogeneous space for E, and thus E ≅ J_{C}.

Our algorithm for the jacobian of (V.4) must therefore produce an answer isomorphic to the E given above, and we will now verify this fact. We will also relate these models for J_{C} to what is found in [AKM^{+}01].

V.7b. Finding the jacobian by the algorithm

To find the jacobian J_{C} of (V.4), we could, in principle, repeat the earlier work in this chapter. (In fact, the computer system Singular is capable of performing calculations over function fields that are finitely generated over either Q or a finite field of small characteristic—for example, line 1 of the code in table V.2 initializes a polynomial ring over the function field Q(a_{ij}). Thus, for suitable k, we could view C as a curve over k(a, b, c, m), where the symbols a, b, c, m are algebraically independent over k. But an attempt to run the code from tables V.2 and V.3, with obvious modifications for the present situation, exhausted our patience.)

We will take a different approach. By glancing at our work in §V.2, we go ahead and form the field extension K = k(ζ_{3}, \sqrt[3]{a}, \sqrt[3]{b}, \sqrt[3]{c}), and then guess that the two matrices

M = [\begin{matrix}1 & 0 & 0 \\ 0 & ζ_{3} & 0 \\ 0 & 0 & ζ_{3}^{2} \end{matrix}] and N = [\begin{matrix}0 & \sqrt[3]{b/a} & 0 \\ 0 & 0 & \sqrt[3]{c/b} \\ \sqrt[3]{a/c} & 0 & 0 \end{matrix}] .

will generate H_{3}. Indeed, this is the case: it is easy to verify that they preserve C, have no fixpoints on C, and their commutator is ζ_{3}.

The Molien series Φ_{H_{3}}(t) is the same as in §V.4, and carrying out a similar procedure as given there (we will omit the details), we choose the primary invariants

P_{1} := xyz, P_{2} := ax^{3} + by^{3} + cz^{3}, P_{3} := abx^{3}y^{3} + acx^{3}z^{3} + bcy^{3}z^{3},

and the secondary invariant P_{4} := ab^{2}x^{3}y^{6} + a^{2}cx^{6}z^{3} + bc^{2}y^{3}z^{6}, and obtain

K [x, y, z]^{H_{3}(K )} ≅ \frac{K [s_{1}, s_{2}, s_{3}, s_{4}]}{⟨9a^{2}b^{2}c^{2}s_{1}^{6}+abcs_{1}^{3}s_{2}^{3}−6abcs_{1}^{3}s_{2}s_{3}+s_{3}^{3}+3abcs_{1}^{3}s_{4}−s_{2}s_{3}s_{4}+s_{4}^{2}⟩}.

Working modulo the ideal of C corresponds to s_{2} = −ms_{1}, and we obtain

J_{C} ≅ Proj \frac{k[s_{1}, s_{3}, s_{4}]}{⟨abc(9abc−m^{3})s_{1}^{6}+6abcms_{1}^{4}s_{3}+s_{3}^{3}+3abcs_{1}^{3}s_{4}+ms_{1}s_{3}s_{4}+s_{4}^{2}⟩}.

We find the points of hyperosculation on C, as in §V.3, by intersecting the hyperplanes fixed by M or by N with C. For example, we find the point [0 : 1 : \sqrt[3]{−b/c}]. Substituting the points of hyperosculation into [P_{1} :: P_{3} :: P_{4}] (cf. the end of §V.4) gives us [0 :: −b^{2} :: b^{3}], [0 :: −c^{2} :: c^{3}], [0 :: −a^{2} :: a^{3}], which are all just [0 :: −1 :: 1].

In summary, the jacobian of (V.4) is

J_{C} ≅ Proj \frac{k[r, s, t]}{⟨abc(9abc − m^{3})r^{6} + 6abcmr^{4}s + s^{3} + 3abcr^{3}t + mrst + t^{2}⟩},  (V.6)

where variables r, s, t have degrees (1; 2; 3), and the origin of the group law is the k-rational point [0 :: −1 :: 1]. We obtained J_{C} as an elliptic curve of degree 6 in the weighted projective plane P_{k}^{(1;2;3)} = Proj(k[r, s, t]). To obtain a non-weighted equation, we now proceed as in §V.6b. The affine patch

Spec \frac{k[u, v]}{⟨v^{2} + muv + 3abcv + u^{3} + 6abcmu + abc(9abc − m^{3})⟩} ⊂ A_{k}^{2}  (V.7)

gives us a Weierstrass model for J_{C}.

V.7c. Applying the Riemann–Roch algorithm

We will now apply the Riemann–Roch algorithm (which we learned from an example in [Con92]) to put the elliptic curve (V.5) into Weierstrass form, so that we may compare it with (V.7).

We must find functions ℘_{x}, ℘_{y} with poles of order 2, 3 at P = [1 : −1 : 0] and no poles elsewhere on E: x^{3} + y^{3} + abcz^{3} + mxyz = 0. We go to the affine patch where x ≠ 0. Set Y = y/x, Z = z/x. Then we are looking at the point P = (−1, 0) on E: 1 + Y^{3} + abcZ^{3} + mY Z = 0. We determine Z to be a uniformizer at (−1, 0), and 1/Z to be a uniformizer at the three points on E where x = 0. Thus we must have

℘_{x} = \frac{quadratic polynomial in Y and Z}{Z^{2}} , ℘_{y} = \frac{cubic polynomial in Y and Z}{Z^{3}} .

The bound on the numerator degree comes from looking at the pole behavior of Y at points where x = 0: the denominator Z^{2} contributes a zero of order 2 at those points, so the numerator can contribute a pole of order at most 2, since the end result should be regular at those points.

Since L(2P) = ⟨1, ℘_{x}⟩, we can adjust ℘_{x} by a scalar multiple of 1 = Z^{2}/Z^{2}; that is, the coefficient of Z^{2} in the numerator of ℘_{x} can be taken to be 0. We also need the numerator of ℘_{x} to not vanish at P, yet vanish at the other two points Q_{1}, Q_{2} on E where Z vanishes. These conditions immediately give some relations on the coefficients of the numerator of ℘_{x}, and tell us the Y^{2} coefficient is nonzero. Scaling, we assume it to be 1. If we then use the equation of E expressed locally at Q_{1}, Q_{2}, we can expand the numerator of ℘_{x} in a series in Z to obtain the condition for the numerator to vanish twice at those two points. We ultimately determine:

℘_{x} = \frac{Y^{2} + \frac{m}{3}Y Z − Y + \frac{m}{3}Z + 1}{Z^{2}}.

As for ℘_{y}, the equation of E itself allows us to eliminate the Y^{3} term from the numerator, and L(3P) = ⟨1, ℘_{x}, ℘_{y}⟩ tells us we can eliminate the Z^{3} term and the Y^{2}Z term (by subtracting off a multiple of ℘_{x}). Repeating the previous work, this time requiring triple vanishing at Q_{1}, Q_{2}, we obtain:

℘_{y} = \frac{Y^{2} + \frac{m}{3}Y Z − Y + \frac{m}{3}Z + 1 + \frac{m^{2}}{9} Z^{2}}{Z^{3}} .

Riemann–Roch tells us that ℓ(6P) = 6, yet certainly { 1, ℘_{x}, ℘_{y}, ℘_{x}^{2}, ℘_{x}℘_{y}, ℘_{x}^{3}, ℘_{y}^{2} } all lie in L(6P), and by comparing with L(5P), we conclude that we can express ℘_{y}^{2} as a linear combination of the rest. That is, there exist a_{i} satisfying:

℘_{y}^{2} + a_{1}℘_{x}℘_{y} + a_{3}℘_{y} = a_{0}℘_{x}^{3} + a_{2}℘_{x}^{2} + a_{4}℘_{x} + a_{6}.

This equation must hold identically on E, so if we expand this using our expressions for ℘_{x}, ℘_{y} and use the equation of E repeatedly to eliminate all terms containing Y^{3}, then the coefficients we end up with must all be 0. From this we easily determine:

℘_{y}^{2} − \frac{m}{3}℘_{x}℘_{y} + abc℘_{y} = \frac{1}{3}℘_{x}^{3} + \frac{abcm}{3}℘_{x} + \frac{abc}{81}(m^{3} − 27abc).

This is E in Weierstrass form. We eliminate the denominators with a linear change-of-variables to obtain

E: y^{2} − mxy + 9abcy = x^{3} + 9abcmx + (abcm^{3} − 27a^{2}b^{2}c^{2}).  (V.8)

If we now apply the substitution x [figure] u and y [figure] (v − 3abc), then we obtain (V.7).

V.7d. Comparing with result from classical invariant theory

If, in addition to our standing assumption char(k) ≠ 3, we also assume char(k) ≠ 2, then we can compare (V.7) with what we would obtain from the formulas in [AKM^{+}01]. There it is shown that J_{C} has equation

y^{2} = 4x^{3} + 108Sx − 27T,  (V.9)

where

S = \frac{1}{6}abcm − \frac{1}{6^{4}} m^{4}, and T = a^{2}b^{2}c^{2} − \frac{20}{6^{3}}abcm^{3} − \frac{8}{6^{6}} m^{6}.

If we scale the linear term of (V.9) by 6^{4}, and the constant term by 6^{6} (cf. [Sil99, §III.1]), then we obtain

y^{2} = 4x^{3} + 108(216abcm − m^{4})x − 216(5832a^{2}b^{2}c^{2} − 540abcm^{3} − m^{6}).  (V.10)

If we instead start with (V.7), and complete the square and the cube (cf. [Sil99, §III.1]), then after cleaning up the denominators we also end up with (V.10).

CHAPTER VI

Example: a pair of quadrics

In the previous chapter, we worked through an example of using our algorithm in the case n = 3. In this chapter, we work through an example for n = 4 far enough to find the Heisenberg group H_{4} and the points of hyperosculation, which demonstrates the following points:

• The ideal of our curve is no longer principal. Thus, whereas before with I = ⟨F ⟩ we put the condition F (Ax) ≡ 0 modulo F (x) to find matrices that preserve the curve, now we must find a Gröbner basis 𝔊 for I, and for each generator F of I, we require F (Ax)/𝔊 ≡ 0, where ·/𝔊 denotes the canonical form obtained by reducing modulo 𝔊.

• Since the parity of n is different, we must use a different method for finding the points of hyperosculation (cf. §IV.5).

• In the previous chapter’s example, the Heisenberg group H_{4} admitted a Schrödinger-like representation in the given coordinate system. That will not be the case presently.

In contrast to the previous chapter, this time around we will rely more on the computer, and we present computer code that, although more abstract, has the advantage of being easily adapted to different values of n.

VI.1. The curve

Let C be the intersection of the two quadrics in P_{Q}^{3} defined by the matrices

[\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] and [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \end{matrix}] .

That is,

C = Proj \frac{Q[w, x, y, z]}{⟨w^{2} + x^{2} + y^{2} + z^{2}, w^{2} + 2x^{2} + 3y^{2} + 4z^{2}⟩}.

Note that C has no Q-rational point: it does not even have a point over R. We will now apply our algorithm to find the points of hyperosculation, the hyperplane configuration, and the jacobian J_{C}.

The code in table VI.1 confirms that C is a smooth degree 4 curve in P_{Q}^{3} of genus 1. It is non-degenerate because none of the ideal generators is linear.

          1  n=4
          2  nMinusOne=n-1
          3  ringPnMinusOne = QQ[w_0..w_nMinusOne]
          4  f1=w_0^2+w_1^2+w_2^2+w_3^2
          5  f2=w_0^2+2*w_1^2+3*w_2^2+4*w_3^2
          6  idealC = ideal(f1,f2)
          7  ringC = ringPnMinusOne / idealC
          8  C = Proj ringC
          9  dim C
        10  codim singularLocus idealC
        11  degree C
        12  HH^1 OO_C

Table VI.1. This Macaulay 2 code vets the input.

Remark. We didn’t really need the computer: since the two matrices are diagonal, and their λequation (as in (III.9)) has distinct roots, the two quadrics intersect in a smooth curve of genus 1 (see [Eis95, §18.3, Example, p.463]).

The code in table VI.2 works out the conditions for a matrix

A = [\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ a_{5} & a_{6} & a_{7} & a_{8} \\ a_{9} & a_{10} & a_{11} & a_{12} \\ a_{13} & a_{14} & a_{15} & a_{16} \end{matrix}]

to preserve the curve, have determinant 1, and satisfy A^{n} = −1. (Thus, by §IV.4b, we are describing potential generators of H_{4}, not H_{4} itself.) Note that, for sake of speed, the code uses a nonelimination term order for the initial computations, and then uses a Gröbner basis conversion process (cf. [CLO98, §2.3]) to obtain a description of the ideal in elimination term order. We end up with the following conditions:

a_{16}^{5} + a_{16} = 0,  a_{15}a_{16} = 0,  a_{14}a_{16} = 0,  a_{14}a_{15} = 0,
3a_{15}^{5} + a_{15} = 0,  a_{13}a_{16} = 0,  a_{13}a_{15} = 0,  a_{13}a_{14} = 0,
3a_{14}^{5} − a_{14} = 0,  a_{12}a_{16} = 0,  a_{12}a_{14} = 0,  a_{12}a_{13} = 0,
a_{13}^{4} − 3a_{14}^{4} + 3a_{15}^{4} + a_{16}^{4} + 1 = 0,  a_{11}a_{15} = 0,  a_{11}a_{14} = 0,  a_{11}a_{13} = 0,
3a_{12}a_{15}^{4} + a_{12} = 0,  a_{11}a_{12} = 0,  a_{10}a_{16} = 0,  a_{10}a_{15} = 0,
a_{12}^{2} − 3a_{15}^{2} = 0,  a_{10}a_{14} = 0,  a_{10}a_{12} = 0,  a_{10}a_{11} = 0,
a_{11}a_{16}^{4} + a_{11} = 0,  a_{9}a_{16} = 0,  a_{9}a_{15} = 0,  a_{9}a_{13} = 0,
a_{11}^{2} − a_{16}^{2} = 0,  a_{9}a_{12} = 0,  a_{9}a_{11} = 0,  a_{9}a_{10} = 0,
a_{10}^{2} − a_{13}^{2} = 0,  a_{8}a_{16} = 0,  a_{8}a_{15} = 0,  a_{8}a_{13} = 0,
3a_{9}a_{14}^{4} − a_{9} = 0,  a_{8}a_{12} = 0,  a_{8}a_{11} = 0,  a_{8}a_{10} = 0,
a_{9}^{2} − 3a_{14}^{2} = 0,  a_{7}a_{16} = 0,  a_{7}a_{15} = 0,  a_{7}a_{14} = 0,
3a_{8}a_{14}^{4} − a_{8} = 0,  a_{7}a_{12} = 0,  a_{7}a_{11} = 0,  a_{7}a_{9} = 0,
a_{8}^{2} + 3a_{14}^{2} = 0,  a_{7}a_{8} = 0,  a_{6}a_{15} = 0,  a_{6}a_{14} = 0,
a_{7}^{2} − a_{13}^{2} = 0,  a_{6}a_{13} = 0,  a_{6}a_{12} = 0,  a_{6}a_{10} = 0,
a_{6}a_{16}^{4} + a_{6} = 0,  a_{6}a_{9} = 0,  a_{6}a_{8} = 0,  a_{6}a_{7} = 0,
a_{6}^{2} − a_{16}^{2} = 0,  a_{5}a_{16} = 0,  a_{5}a_{14} = 0,  a_{5}a_{13} = 0,
3a_{5}a_{15}^{4} + a_{5} = 0,  a_{5}a_{11} = 0,  a_{5}a_{10} = 0,  a_{5}a_{9} = 0,
a_{5}^{2} + 3a_{15}^{2} = 0,  a_{5}a_{8} = 0,  a_{5}a_{7} = 0,  a_{5}a_{6} = 0.
a_{4} − a_{7}a_{10}a_{13}^{3} = 0,
a_{3} + a_{8}a_{9}a_{14}^{3} = 0,
a_{2} + a_{5}a_{12}a_{15}^{3} = 0,
a_{1} − a_{6}a_{11}a_{16}^{3} = 0,

64 VI. Example: a pair of quadrics

 1  // Lines marked ``//[**]'' are specific to the parity of n,
 2  // or to the particular value of n, or to the particular equations
 3  // being used in this example. Adjust these lines as necessary.
 4
 5  int n = 4; //[**]
 6
 7  ring ring_PnMinusOne = 0,(x(0..(n-1))),dp;
 8  poly f1 = x(0)^2+x(1)^2+x(2)^2+x(3)^2; //[**]
 9  poly f2 = x(0)^2+2*x(1)^2+3*x(2)^2+4*x(3)^2; //[**]
10  ideal ideal_C = f1,f2; //[**]
11
12  option(redSB); // Now ``groebner'' will reduce result...
13  ideal_C = groebner(ideal_C);
14
15  int n_squared = n*n;
16  ring ring_combined = (0,a(1..n_squared)),(x(0..(n-1))),dp;
17  ideal ideal_C_combined = imap(ring_PnMinusOne,ideal_C);
18  // Singular must be reminded it is groebner:
19  ideal_C_combined = groebner(ideal_C_combined);
20  matrix A[n][n] = a(1..(n_squared));
21  matrix vars[n][1] = x(0..(n-1));
22  matrix newvars = A * vars;
23  newvars;
24  map map_A = ring_combined,newvars[1,1],newvars[2,1],newvars[3,1],newvars[4,1]; //[**]
25
26  ideal_C_combined;
27
28  poly varprod = 1;
29  int i;
30  for (i = 0; i < n; i++) {
31    varprod = varprod * x(i);
32    }
33
34  ring ring_coefficients = 0,(a(1..(n_squared))),dp;
35  ideal ideal_conditions;
36
37  matrix AA = imap(ring_combined,A);
38  poly determinant_one = 1 - det(AA);
39  ideal_conditions = ideal_conditions + determinant_one;
40
41  // Extra (optional) conditions: (n)th power should be +1 or -1.
42  // Note: For n even, we won't get a group, but we'll find generators...
43  LIB "matrix.lib"; // Lets us call power(), unitmat(), ...
44  matrix AA_nth_power = power(AA,n) + (-1)^n * unitmat(n);
45  int j;
46  for (i = 1; i <= n; i++) {
47    for (j = 1; j <= n; j++) {
48      ideal_conditions = ideal_conditions + AA_nth_power[i,j];
49      }
50    }
51
52  setring ring_combined;
53  int eqn_count = ncols(ideal_C_combined);
54  for (i = 1; i <= eqn_count; i++) {
55    setring ring_combined;
56    poly generator = ideal_C_combined[i];
57    poly mapped_generator = map_A(generator);
58    mapped_generator;
59    poly reduced = reduce(mapped_generator, ideal_C_combined);
60    reduced;
61    matrix coefficients = coef(reduced,varprod);
62    int coef_count = ncols(coefficients);
63    for (j = 1; j <= coef_count; j++) {
64      setring ring_combined;
65      poly coefficient = coefficients[2,j];
66      setring ring_coefficients;
67      poly mapped_coefficient = imap(ring_combined,coefficient);
68      mapped_coefficient = cleardenom(mapped_coefficient);
69      ideal_conditions = ideal_conditions + mapped_coefficient;
70      }
71    }
72
73  setring ring_coefficients;
74  ideal_conditions = groebner(ideal_conditions);
75  ring ring_coefficients_lex = 0,(a(1..(n_squared))),lp;
76  ideal ideal_conditions_lex = fglm(ring_coefficients,ideal_conditions);
77  ideal_conditions_lex;

Table VI.2. This Singular code find equations describing generators of H_{4}.

To solve the system, let K = Q(ζ_{8}, \sqrt[4]{3}), where ζ_{8} is a symbol satisfying ζ_{8}^{4} = −1 and \sqrt[4]{3} is a symbol satisfying (\sqrt[4]{3})^{4} = 3. Modulo the 4th roots of unity \{ 1, ζ_{8}^{2}, ζ_{8}^{4}, ζ_{8}^{6} \}, we find 12 solutions (the signs must be assigned so that the determinant is 1):

[\begin{matrix}0 & 0 & 0 & ±ζ_{8} \\ 0 & 0 & ±ζ_{8} & 0 \\ 0 & ±ζ_{8} & 0 & 0 \\ ζ_{8} & 0 & 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 & ±ζ_{8}^{2}/\sqrt[4]{3} & 0 \\ 0 & 0 & 0 & ±ζ_{8}^{2} \sqrt[4]{3} \\ ±\sqrt[4]{3} & 0 & 0 & 0 \\ 0 & 1/\sqrt[4]{3} & 0 & 0 \end{matrix}] ,

[\begin{matrix}0 & ±ζ_{8}^{3}/\sqrt[4]{3} & 0 & 0 \\ ±ζ_{8}^{3} \sqrt[4]{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & ±ζ_{8} \sqrt[4]{3} \\ 0 & 0 & ζ_{8}/\sqrt[4]{3} & 0 \end{matrix}] .

Their squares are the elements of order 2 in PGL_{4} (the signs must be assigned so that the determinant is 1 and so that the matrix is not a scalar):

[\begin{matrix}±ζ_{8}^{2} & 0 & 0 & 0 \\ 0 & ±ζ_{8}^{2} & 0 & 0 \\ 0 & 0 & ±ζ_{8}^{2} & 0 \\ 0 & 0 & 0 & ζ_{8}^{2} \end{matrix}] .

We arbitrarily choose two matrices of order 4 whose commutator is a primitive 4th root of unity:

M = [\begin{matrix}0 & 0 & 0 & −ζ_{8} \\ 0 & 0 & ζ_{8} & 0 \\ 0 & ζ_{8} & 0 & 0 \\ ζ_{8} & 0 & 0 & 0 \end{matrix}] , N = [\begin{matrix}0 & 0 & ζ_{8}^{2}/\sqrt[4]{3} & 0 \\ 0 & 0 & 0 & ζ_{8}^{2} \sqrt[4]{3} \\ −\sqrt[4]{3} & 0 & 0 & 0 \\ 0 & 1/\sqrt[4]{3} & 0 & 0 \end{matrix}] .

We have M N M^{−1}N^{−1} = ζ_{8}^{2}. The Heisenberg group is

H_{4} = { ζ_{8}^{2a}M^{b}N^{c} : a, b, c ∈ Z/4Z }.

VI.2. Finding the [−1] matrices

By running code similar to before, but replacing the condition A^{4} = −1 with A^{2} = 1 and also trying again (cf. §IV.4c) with A^{2} = ζ_{8}^{2} or simply A^{4} = −1, and then checking that each matrix has 4 fixpoints on C, we find the following 16 matrices (the signs must be assigned so that the determinant is 1):

[\begin{matrix}0 & 0 & ±ζ_{8}^{3}/\sqrt[4]{3} & 0 \\ 0 & 0 & 0 & ±ζ_{8}^{3} \sqrt[4]{3} \\ ±ζ_{8} \sqrt[4]{3} & 0 & 0 & 0 \\ 0 & ζ_{8}/\sqrt[4]{3} & 0 & 0 \end{matrix}] , [\begin{matrix}0 & ±ζ_{8}^{2}/\sqrt[4]{3} & 0 & 0 \\ ±ζ_{8}^{2} \sqrt[4]{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & ±\sqrt[4]{3} \\ 0 & 0 & 1/\sqrt[4]{3} & 0 \end{matrix}] ,

[\begin{matrix}0 & 0 & 0 & ±ζ_{8}^{2} \\ 0 & 0 & ±ζ_{8}^{2} & 0 \\ 0 & ±ζ_{8}^{2} & 0 & 0 \\ ζ_{8}^{2} & 0 & 0 & 0 \end{matrix}] , [\begin{matrix}±ζ_{8} & 0 & 0 & 0 \\ 0 & ±ζ_{8} & 0 & 0 \\ 0 & 0 & ±ζ_{8} & 0 \\ 0 & 0 & 0 & ζ_{8} \end{matrix}] .

VI.3. Finding a point of hyperosculation

Since n is even, the points of hyperosculation are the 16 fixpoints of 4 of the [−1] matrices (cf. §III.9a). We simply take each matrix, look for a fixpoint, look for the osculating hyperplane at that fixpoint, and check whether it hyperosculates; if not, we move on to the next matrix.

Let us look at the matrix

T = [\begin{matrix}−ζ_{8} & 0 & 0 & 0 \\ 0 & ζ_{8} & 0 & 0 \\ 0 & 0 & ζ_{8} & 0 \\ 0 & 0 & 0 & ζ_{8} \end{matrix}] .

It fixes the isolated point [1 : 0 : 0 : 0] (which is not on C) and fixes every point in the eigenspace spanned by [0 : 1 : 0 : 0], [0 : 0 : 1 : 0], and [0 : 0 : 0 : 1]. That eigenspace is just the hyperplane w = 0, which intersects C in the 4 points

[0 : 1 : ±\sqrt{−2} : ±1].

Let

O = [0 : 1 : \sqrt{−2} : 1].

It is easy to see that the hyperosculating hyperplane at O (assuming it exists) must be fixed by T. The hyperplanes fixed by T correspond to the eigenvectors of T^{t}. We determine that T fixes the isolated hyperplane w = 0 (which we just saw cannot be the hyperosculating hyperplane at O), and T also fixes all the hyperplanes spanned by x = 0, y = 0, z = 0 (these are all the hyperplanes containing the point [1 : 0 : 0 : 0]).

We thus look at hyperplanes corresponding to ℓ = [0 : ∗ : ∗ : ∗]^{t} that go through O, i.e., the hyperplanes sx + ty − (1 + t\sqrt{−2})z = 0, where [s : t] ∈ P_{Q}^{1}(Q̄). We look for [s : t] values that maximize the number of times the hyperplane meets the curve at O. We eventually discover, for [s : t] = [1 : 2\sqrt{−2}], the hyperplane meets C only at O. Thus O is a point of hyperosculation.

VI.4. The invariant theory of H_{4}

To eventually describe

J_{C} ≅ Proj \frac{Q[w, x, y, z] ∩ K[w, x, y, z]^{H_{4}(K)}}{I ∩ K[w, x, y, z]^{H_{4}(K)}} ,

which we will not do here, we would first need to get our hands on K[w, x, y, z]^{H_{4}(K)}. We will now indicate how that calculation commences. The Molien series

Φ_{H_{4}}(t) = \frac{1}{64} ∑_{τ∈H_{4}(K)} \frac{1}{det(id − tτ)}

= \frac{1 − t^{4} + 8t^{2} − t^{12} + t^{16}}{(t^{4} − 1)^{4}(t^{4} + 1)}

= 1 + 2t^{4} + 12t^{8} + 29t^{12} + 63t^{16} + 112t^{20} + 186t^{24} + 283t^{28} + …

tells us there are two linearly independent invariants in degree 4, twelve in degree 8, and so on. We can verify that the following four polynomials,

\begin{aligned}P_{1} & = wxyz, \\ P_{2} & = 3w^{4} − x^{4} + y^{4} − 3z^{4}, \\ P_{3} & = x^{4}y^{4} + 9w^{4}z^{4}, \\ P_{4} & = (x^{2}y^{2} + 3w^{2}z^{2})(w^{2}y^{2} + x^{2}z^{2}), \end{aligned}

are algebraically independent invariants with (0, 0, 0, 0) being the only solution to P_{1} = 0, P_{2} = 0, P_{3} = 0, and P_{4} = 0. Thus they serve as primary invariants, and we rewrite the Molien series in the form

Φ_{H_{4}}(t) = \frac{1 + 7t^{8} + 7t^{12} + t^{20}}{(1 − t^{4})^{2}(1 − t^{8})^{2}} .

We see that there are 7 secondary invariants in degree 8, 7 in degree 12, and 1 in degree 20. We have not carried out the calculation to find those secondary invariants.

APPENDIX A

Facts about curves of genus 1

In this appendix, we review facts concerning k-curves of genus 1, where k is a perfect field. Our results are mostly obtained from the theory of elliptic curves, as presented in [Sil99], by separating an elliptic curve into two aspects: the underlying curve of genus 1, and the curve’s jacobian.

A.1. Consequences of Riemann–Roch on curves of genus 1

We recall the Riemann–Roch theorem for a curve of arbitrary genus.

Theorem (Riemann–Roch). Let X be a k-curve of genus g, let K be a canonical divisor, and let D be a k-rational divisor on X. Let ℓ_{k}(·) = dim_{k} L_{k}(·) = dim_{k} H^{0}(X, O(·)). Then

ℓ_{k}(D) − ℓ_{k}(K − D) = deg(D) + 1 − g.

Proof. Combine [Har77, IV.1.3] with [Sil99, II.5.8.1]). □

We now return to the context of C being a curve of genus 1, and explore the consequences of the above theorem.

A.1a. The canonical divisor and differential forms

Substituting D = 0 and D = K into Riemann–Roch gives

ℓ_{k}(K) = 1 and deg(K) = 0, whence K ∼ 0.

In words: there exist global differential forms, any two differ by a scalar multiple, and they are nowhere vanishing.

Any choice of nonzero global differential form is usually called “the invariant differential”, referring to the fact that they are invariant under translation by the action of the jacobian J_{C} on C; this follows from [Sil99, III.5.1].

A.1b. Dimension of complete linear systems

Corollary (Riemann–Roch on curves of genus 1). Let C be a k-curve of genus 1, and D a k-rational divisor. Then:

ℓ_{k}(D) − ℓ_{k}(−D) = deg(D).

The relationship between the degree and dimension may be summarized as follows. The first three statements hold for all curves, the last two for curves of genus 1:

• If deg(D) < 0, then ℓ_{k}(D) = 0.

• If deg(D) = 0 and D ≁ 0, then ℓ_{k}(D) = 0.

• If deg(D) = 0 and D ∼ 0, then ℓ_{k}(D) = 1.

• If deg(D) > 0, then ℓ_{k}(D) = deg(D).

• If D is effective, then ℓ_{k}(D) ≥ 1.

A.1c. No two points are linearly equivalent

Let P, Q ∈ C(k̄) be two distinct points. If we had P ∼ Q, then P − Q would be the divisor of a function, implying the existence of a non-constant function in L(Q), contradicting ℓk̄(Q) = 1.

A.1d. Effective representatives

For D a k-rational divisor, the fact

deg(D) > 0 [figure] ℓ_{k}(D) = deg(D)

tells us that the complete linear system |D|_{k} (which comprises effective k-rational divisors linearly equivalent to D) is nonempty whenever deg(D) > 0. In other words: if D has positive degree, then the class [D] has an effective k-rational representative.

Thus, if D has degree 1, then the class [D] is represented by a k-rational point. But, by A.1c, no two distinct points are linearly equivalent. Thus: if D has degree 1, then the class [D] has a unique k-rational point as representative.

The last statement is true more generally. We no longer require D to be k-rational, but we do require the class [D] to be k-rational, which means D^{σ} ∼ D for all Galois automorphisms σ. Temporarily taking our ground field to be k̄, the previous paragraph tells us that [D] has a unique k̄-rational point P as representative. But then k-rationality of [P] implies P^{σ} ∼ P, and uniqueness forces P^{σ} = P. Thus P is k-rational. In short: each k-rational divisor class of degree 1 has a unique k-rational point as representative.

Warning. It is in general not the case that the divisor class group Cl_{k}(C) comprises k-rational divisors modulo linear equivalence. An element of Cl_{k}(C) is a k-rational class, i.e., a class [D] so that each representative D satisfies D^{σ} ∼ D; such a class need not admit a k-rational representative. However, as we saw above, on a curve of genus 1, each k-rational class of degree 1 does admit a k-rational representative.

A.2. The group law

The set C(k) of k-rational points on C may be empty. When it is nonempty, then choosing any O ∈ C(k) leads to the following composition rule: for P, Q ∈ C(k), define P ⊕ Q to be the unique k-rational point (see A.1d) linearly equivalent to the degree 1 k-rational divisor P + Q − O.

The calculation

(P ⊕ Q) ⊕ R ∼ (P ⊕ Q) + R − O ∼ (P + Q − O) + R − O = (P + Q + R) − 2O

renders moot the order in which P, Q, R appear and the order in which they are combined, showing the binary operation “⊕” to be both commutative and associative. It is furthermore immediate that O is an identity element for this operation. Finally, given P ∈ C(k), let ⊖P be the unique k-rational point linearly equivalent to 2O − P. Then ⊖P is the additive inverse of P:

P ⊕ (⊖P) ∼ P + (⊖P) − O ∼ P + (2O − P) − O = O.

Thus the operation “⊕” defines an abelian group law on the set C(k) with O as identity element.

In summary: when C(k) is nonempty, each choice of O ∈ C(k) leads to an abelian group law on C(k) with O as identity element.

This abstract description of the group law is intrinsic to the curve. The reader may be familiar with the extrinsic “chord-and-tangent law” defined on nonsingular cubic curves in P_{k}^{2}. (A description, along with pictures, appears in practically every book on elliptic curves.) That law has the following description: fix any O ∈ C(k) to be the identity element, and then agree that three points in C(k) sum to O if and only if they are collinear. Not every curve of genus 1 occurs as a cubic in P_{k}^{2}. But for the ones that do, we have the following result:

Theorem A.2.1. The intrinsic and extrinsic laws agree.

Proof. Fix O ∈ C(k). For P, Q ∈ C(k), their intrinsic sum is the unique point linearly equivalent to P + Q − O. Now let “⊕” denote the extrinsic law. If we show P ⊕ Q ∼ P + Q − O, then we have shown the two laws to agree.

Let R ∈ C(k) be such that P, Q, R are collinear. (In other words, take the line through P and Q and find its third point of intersection with the curve. If P = Q, then use the tangent line.) The extrinsic law says P ⊕ Q ⊕ R = O. Thus R is the inverse of P ⊕ Q. Let S ∈ C(k) be such that O, R, S are collinear. Then S must be P ⊕ Q. The two lines define a rational function that gives P + Q + R ∼ O + R + S, whence P + Q − O ∼ S = P ⊕ Q. □

A.2a. Isogenies

When C(k) is nonempty and O ∈ C(k) has been fixed, the pair (C, O) is called an elliptic curve. A curve morphism

(C, O) [figure] (C′, O′) with O [figure] O′

is called an isogeny. The trivial isogeny is the constant map with value O′. Non-trivial isogenies are finite morphisms and thus have a degree. Two elliptic curves are isogenous if there exists a non-trivial isogeny between them.

Theorem A.2.2. Every k-isogeny C [figure] C′ is necessarily a group homomorphism C(k) [figure] C′(k).

Proof. Let P, Q ∈ C(k). Then P ⊕ Q is the unique point linearly equivalent to P + Q − O. Since ϕ_{∗} takes principal divisors to principal divisors, we have ϕ(P ⊕ Q) ∼ ϕ(P) + ϕ(Q) − ϕ(O), whence ϕ(P ⊕ Q) = ϕ(P) ⊕ ϕ(Q). □

Corollary A.2.3. Any k-morphism C [figure]C with a fixpoint O ∈ C(k) is automatically a group endomorphism C(k) [figure] C(k) of the elliptic curve (C, O).

The above result is an ingredient into the structure theorem for morphisms of curves of genus 1. (See A.4.1.)

A.3. The jacobian action

Recall that a k-rational point on the jacobian J_{C} is the same thing as a k-rational divisor class on C of degree 0. In other words, J_{C}(k) = Cl_{k}^{0}(C), and J_{C}(k̄) = Cl0 k̄(C). There is a canonical action of J_{C} on C, whose behavior on k̄-valued points is:

J_{C}(k̄) × C(k̄) [figure] C(k̄),
([Z], P ) [figure] unique point linearly equivalent to [Z + P].

Proposition A.3.1. The action of J_{C} on C is simply transitive (and thus faithful and fixpoint-free).

Proof. Let P, Q ∈ C(k̄). The action is transitive since [Q − P] ∈ J_{C}(k̄) sends P to Q. Let [Z] ∈ J_{C}(k̄) also have that property. Then Z + P ∼ Q, whence [Z] = [Q − P]. □

Thus J_{C} comprises a part of the group of curve automorphisms of C. The precise structure of that group is explained in A.4c.

A.3a. Torsion packets on curves of genus 1

Since J_{C} acts on C, for each n ≥ 1 also J_{C}[n] acts on C, where J_{C}[n] is the kernel of multiplicationby-n:

J_{C}[n](k̄) = J_{C}(k̄)[n].

Definition. An n-torsion packet on C is a collection of points in C(k̄) that comprises one orbit under the action of J_{C}[n](k̄).

Thus we see: the set of n-torsion packets is in one-to-one correspondence with (C/J_{C}[n])(k̄).

If P, Q ∈ C(k̄) lie in the same n-torsion packet, then P − Q defines a class in J_{C}(k̄)[n], whence nP ∼ nQ. Thus another description: an n-torsion packet is a maximal collection of points in C(k̄) so that if P, Q are any two of them, then nP ∼ nQ.

Another way to think about n-torsion packets is as follows. Let T ⊂ C(k̄) be an n-torsion packet. Choosing any O ∈ T, we get the elliptic curve (Ck̄, O). Then T is simply the n-torsion on that curve, i.e., T = (Ck̄, O)[n](k̄), and the other n-torsion packets on C are the cosets of T in (Ck̄, O)(k̄). In summary, talking about an n-torsion packet on C is tantamount to saying, “here is a collection of points which would be the n-torsion were we to choose one of them as group law origin”.

Proposition A.3.2. The size of each n-torsion packet is the constant |J_{C}[n](k̄)|. In particular, if char(k) ∤ n, then each n-torsion packet comprises n^{2} distinct elements of C(k̄).

Proof. The first statement follows from the action of J_{C} on C being simply transitive. The hypothesis char(k) ∤ n guarantees that J_{C}(k̄)[n] has order n^{2}. (As a group, it is isomorphic to Z/nZ×Z/nZ, but as a Gal(k̄/k)-module, its structure can be more complicated.) □

Proposition A.3.3. The Galois conjugate of an n-torsion packet is again an n-torsion packet. In other words, each n-torsion packet is either permuted among itself or swapped as a whole with some other n-torsion packet.

Proof. Any point in an n-torsion packet determines the packet by taking that point’s orbit under the action of J_{C}[n]. Since J_{C}[n] and the action of J_{C} on C are both defined over k, what happens to an n-torsion packet is entirely determined by what happens to one of its representative points. Such a point either stays within the packet or moves to a different packet. The rest of the packet follows along. □

A.4. Morphisms

A.4a. Morphisms between curves of genus 1

In A.3, we saw that J_{C} acts simply transitively on C. Thus each [Z] ∈ J_{C}(k̄) gives rise to a distinct fixpoint-free curve automorphism of C. Applying this to two curves C and C′ of genus 1, we obtain the following description of Hom(Ck̄, C′ ; k̄).

Theorem A.4.1. Fix choices of O ∈ C(k̄) and O′ ∈ C′(k̄). Then each curve morphism Ck̄ [figure] C′ ; k̄ can be decomposed uniquely in the form τ ∘ ϕ, where τ ∈ J_{C′}(k̄) and ϕ ∈ Hom((Ck̄, O), (C′ ; k̄, O′)). In other words, every morphism is a homomorphism followed by a translation.

Proof. Call the morphism η. For existence of the decomposition, write η = [η(O) − O′] ∘ ([O′ − η(O)] ∘ η). Since [O′ − η(O)] ∘ η sends O [figure]O′, it is a homomorphism from the elliptic curve (C, O) to the elliptic curve (C′, O′). (That was A.2.2.) For uniqueness, say τ ∘ ϕ = τ′ ∘ ϕ′. Then ϕ = τ^{−1} ∘ τ′ ∘ ϕ′. Thus τ^{−1} ∘ τ′ fixes O′. By the simple transitivity of the J_{C′}-action, we must have τ = τ′, whence also ϕ = ϕ′. □

Remark. Instead of post-translating on C′ after applying ϕ, one might hope to instead pre-translate on C. That works only if the morphism C [figure] C′ is assumed non-constant.

One might ask: what about morphisms C [figure] C′ that are k-rational? Is it necessarily the case that ϕ and τ in the theorem will also be k-rational? The answer is no: we have τ ∘ϕ = (τ ∘ϕ)^{σ} = τ^{σ} ∘ϕ^{σ}, but ϕ^{σ} need not take O [figure] O′. But if O and O′ are k-rational, then ϕ^{σ} does take O [figure] O′, whence ϕ = ϕ^{σ} and τ = τ^{σ}. We have proved:

Theorem. Fix choices of O ∈ C(k) and O′ ∈ C′(k), assuming they exist. Then each k-rational curve morphism C [figure] C′ can be decomposed uniquely in the form τ ∘ ϕ, where τ ∈ J_{C′}(k) and ϕ ∈ Hom_{k}((C, O), (C′, O′)). In other words, every k-rational morphism is a k-rational homomorphism followed by a k-rational translation.

A.4b. Endomorphisms of curves of genus 1

By the above results, we obtain the following description of morphisms Ck̄ [figure] Ck̄, i.e., of End(Ck̄).

Theorem A.4.2. Fix a choice of O ∈ C(k̄). Then each curve morphism Ck̄ [figure] Ck̄ can be decomposed uniquely in the form τ ∘ ϕ, where ϕ ∈ End(Ck̄, O) and τ ∈ J_{C}(k̄).

Proof. Follows immediately from A.4.1. □

Remark. It would be nice to better elucidate the structure of the monoid End(Ck̄) as some kind of product between End(Ck̄, O) and J_{C}(k̄). Unfortunately, the composition law appears to have no simple description. Given τϕ and τ′ϕ′, how does one write τϕτ′ϕ in the desired form? As we’ll see in the next section, when we restrict attention to automorphisms, we’ll obtain a semidirect product.

Concerning k-rational morphisms C [figure] C, similar to before we can say:

Theorem. Fix a choice of O ∈ C(k), assuming one exists. Then each k-rational curve morphism C [figure] C can be decomposed uniquely in the form τ ∘ ϕ, where τ ∈ J_{C}(k) and ϕ ∈ End_{k}(C, O).

To finish the story, we need to understand the structure of the endomorphism ring End(Ck̄, O). Lying in there are the multiplication-by-n maps [n], thus putting Z ⊆ End(Ck̄, O). When there is more than this, we say that the elliptic curve (Ck̄, O) has complex multiplication. The possibilities are as follows.

Theorem. Fix O ∈ C(k̄). Then End(Ck̄, O) is one of the following:

(1) Z;

(2) an order in the ring of integers of an imaginary quadratic extension of Q;

(3) an order in a definite quaternion algebra over Q. If k is a finite field, then the first possibility is ruled out. If k has characteristic 0, then the third possibility is ruled out.

Proof. See [Sil99, III.9]. □

A morphism Ck̄ [figure] Ck̄ decomposed as above as τ ∘ ϕ is called a pure translation when ϕ = id. Thus J_{C}(k̄) is the group of pure translations of C.

Theorem A.4.3. A non-identity morphism Ck̄ [figure]Ck̄ is fixpoint-free if and only if it is a pure translation.

Proof. That non-identity pure translations are fixpoint-free was explained in section A.3. For the other direction, we will establish the contrapositive. Assume that τ ∘ ϕ is a morphism with ϕ ≠ id. We must show that the morphism has a fixpoint. The elliptic curve endomorphism id − ϕ of (Ck̄, O) is non-constant whence surjective. Furthermore, the map τ corresponds to a translation map on (Ck̄, O), say by the point P. Since id−ϕ is surjective, there exists Q ∈ C(k̄) with Q−ϕ(Q) = P. Then we have (τ ∘ ϕ)(Q) = P + ϕ(Q) = Q, showing τ ∘ ϕ to have a fixpoint. □

Corollary A.4.4. A fixpoint-free morphism Ck̄ [figure] Ck̄ is an automorphism.

Remark. Fixpoints are discussed further in section A.4e.

A.4c. Automorphisms of curves of genus 1

Now we specialize to the case of isomorphisms Ck̄ [figure] Ck̄, i.e., we describe Aut(Ck̄).

Theorem A.4.5. Fix a choice of O ∈ C(k̄). Then the curve automorphism group of Ck̄ is

Aut(Ck̄) = J_{C}(k̄) ⋊ Aut(Ck̄, O).

Proof. The unique decomposition aspect follows immediately from A.4.1. The semidirect product structure is evident by computing the group law: given τ ∘ ϕ and τ′ ∘ ϕ′, their composition τϕτ′ϕ′ may be written τ(ϕτ′ϕ^{−1})ϕϕ′, and now we merely have to convince ourselves that ϕτ′ϕ^{−1} is a pure translation. In other words, we must convince ourselves that J_{C}(k̄) is a normal subgroup of Aut(Ck̄).

Let [Z] ∈ J_{C}(k̄) and ϕ ∈ Aut(Ck̄). The action of [Z] on C(k̄) is to take any point P ∈ C(k̄) to the unique point representing the divisor Z + P. What does ϕ[Z]ϕ^{−1} do? First P goes to ϕ^{−1}(P), then [Z] carries that to the unique point linearly equivalent to Z + ϕ^{−1}(P), and thus ϕ carries it to the unique point linearly equivalent to ϕ_{∗}(Z) + P. In short, ϕ ∘ [Z] ∘ ϕ^{−1} is the same as [ϕ_{∗}Z]. □

Remark A.4.6. As described in the proof, the semidirect product structure tells us how to combine two automorphisms:

([Z], ϕ) ∘ ([Z′], ϕ′) = ([Z] + [ϕ_{∗}Z′], ϕ ∘ ϕ′).

Concerning k-rational isomorphisms C [figure] C, similar to before we can say:

Theorem. Fix a choice of O ∈ C(k), assuming one exists. Then the group of k-rational curve automorphisms of C is:

Aut_{k}(C) = J_{C}(k) ⋊ Aut_{k}(C, O).

To finish the story, we need to understand the structure of the automorphism group Aut(Ck̄, O).

Theorem A.4.7. The structure of Aut(Ck̄, O) depends on the j-invariant of C and the characteristic of k:

Aut(Ck̄, O) = \begin{cases}μ_{2}(k̄), & if j(C) ≠ 0, 12^{3} and char(k) ≠ 2; \\ μ_{4}(k̄), & if j(C) = 12^{3} and char(k) ≠ 2, 3; \\ μ_{6}(k̄), & if j(C) = 0 and char(k) ≠ 2, 3; \\ C_{2}, & if j(C) ≠ 0 (= 12^{3}) and char(k) = 2; \\ C_{4} ⋉ C_{3}, & if j(C) = 0 (= 12^{3}) and char(k) = 3; \\ C_{3} ⋉ Q_{8}, & if j(C) = 0 (= 12^{3}) and char(k) = 2. \end{cases}

Remark. Here C_{n} denotes a cyclic group of order n, and Q_{8} a quaternion group of order 8. We use μ_{n}(k̄) in place of C_{n} in the first three cases, as this furthermore indicates the Galois module structure of the group. But when char(k) = 2, 3, that structure depends on more than just the j-invariant, and is not given here.

Proof. See [Sil99, III.10.1, III.10.2] and the proof of [Sil99, A.1.2c]. □

Corollary A.4.8. Each elliptic curve (Ck̄, O) admits a unique automorphism of order 2 (usually denoted [−1]). Its fixpoints are the points of 2-torsion (Ck̄, O)[2](k̄), and if Q is one of those points, then the same automorphism is also the [−1]-automorphism of the elliptic curve (Ck̄, Q).

Proof. The first statement follows from the theorem. The second statement follows from uniqueness. □

Theorem A.4.9. Fix a choice of O ∈ C(k̄). In Aut(Ck̄) = J_{C}(k̄) ⋊ Aut((Ck̄, O)), the order 2 automorphism of Ck̄ with a fixpoint are precisely the elements of the form τ∘[−1], where τ ∈ J_{C}[n](k̄), and [−1] is the unique order 2 automorphism of the elliptic curve (Ck̄, O).

Proof. By A.4.5, our automorphism is of the form τ ∘ ϕ for some τ ∈ J_{C}(k̄) and some ϕ ∈ Aut((Ck̄, O)). From the semidirect product structure (cf. A.4.6), it is easy to see that τ ∘ ϕ having order 2 implies ϕ itself has order 2. Therefore, by A.4.8, our automorphism is of the form τ ∘ [−1].

It remains to show that every τ ∈ J_{C}(k̄) can occur. In other words, for an arbitrary such τ, we must show that τ ∘ [−1] has order 2 and a fixpoint. Now τ = τ_{P} (translation-by-P) for some P ∈ C(k̄). Let Q ∈ C(k̄) be such that Q ⊕ Q = P, where the group law is on (Ck̄, O). Then (τ ∘ [−1])(Q) = P ⊖ Q = Q, and τ_{P} ∘ [−1] ∘ τ_{P} ∘ [−1] = τ_{P} ∘ ([−1] ∘ τ_{P} ∘ [−1]) ∘ ([−1] ∘ [−1]) = τ_{P} ∘ τ_{⊖P} = id. □

A.4d. Separable and inseparable degree of a morphism

Theorem A.4.10. Let ϕ: C [figure] C′ be a finite morphism of curves of genus 1. Then for all Q ∈ C′(k̄) and all P ∈ C(k̄),

#ϕ^{−1}(Q) = sep deg ϕ,
e_{ϕ}(P) = insep deg ϕ.

Proof. Fix arbitrary O ∈ C(k̄) and set O′ = ϕ(O). Then ϕ is a non-constant isogeny (C, O) [figure] (C′, O′). Now apply [Sil99, III.4.10a]. □

A.4e. The degree and fixpoints of a curve endomorphism

For any morphism η: C [figure] C, let # Fix(η) denote the number of fixpoints of η lying in C(k̄). In A.4b we saw that, after fixing a choice O ∈ C(k̄), each morphism η: C [figure] C has a unique decomposition as an endomorphism of (C, O) followed by a pure translation:

η = τ ∘ ϕ, where ϕ ∈ End(C, O) and τ ∈ J_{C}(k̄).

How this decomposition relates to the existence of fixpoints was given in A.4.3.

Theorem A.4.11. In the above decomposition, we have:

deg(τ ∘ ϕ) = deg(ϕ),

# Fix(τ ∘ ϕ) = \begin{cases}# Fix(ϕ), & if ϕ ≠ id; \\ ∞, & if ϕ = id and τ = id; \\ 0, & if ϕ = id and τ ≠ id. \end{cases} .

Proof. Since pure translations have degree 1, we have deg(τ ∘ ϕ) = deg(ϕ). The rest of the proof concerns fixpoints. Letting “⊖” refer to subtraction on the elliptic curve (C, O), we can certainly say that # Fix(τ ∘ ϕ) is the same as the number of points in C(k̄) sent to O by the map (τ ∘ ϕ) ⊖ id.

Lemma. The map (τ ∘ ϕ) ⊖ id is finite precisely when ϕ ≠ id. Otherwise (τ ∘ ϕ) ⊖ id is the constant map with image τ(O).

Proof. Observe that the action of τ is simply addition by the point τ(O) ∈ C(k̄); that is, τ(P) = P ⊕ τ(O).

If ϕ = id, then (τ ∘ ϕ) ⊖ id takes P [figure]P ⊕ τ(O) ⊖ P, so that (τ ∘ ϕ) ⊖ id is constant with image τ(O).

If ϕ ≠ id, then there exists a P ∈ C(k̄) with P ≠ ϕ(P). Then (τ ∘ ϕ) ⊖ id takes P [figure] ϕ(P) ⊕ τ(O) ⊖ P, while it takes O [figure] τ(O). The two images are distinct precisely because ϕ(P) ⊖ P ≠ O. Thus (τ ∘ ϕ) ⊖ id is non-constant, whence finite. □

We now continue with the proof of the theorem. Agreeing that constant maps have degree 0, the number of points in C(k̄) sent to O by the map (τ ∘ ϕ) ⊖ id is the same as its separable degree so long as the map either is finite or is constant with image away from O. But it is constant with image O only when (τ ∘ ϕ) = id. Thus we have shown:

# Fix(τ ∘ ϕ) = sep deg((τ ∘ ϕ) ⊖ id) if τ ∘ ϕ ≠ id.

By observing that (τ ∘ ϕ) ⊖ id carries P [figure] ϕ(P) ⊕ τ(O) ⊖ P, we see that the “⊕τ(O)” part does not affect the degree, whence

# Fix(τ ∘ ϕ) = sep deg(ϕ ⊖ id) if τ ∘ ϕ ≠ id.

Of course, we also have # Fix(ϕ) = sep deg(ϕ ⊖ id) so long as ϕ ≠ id, and thus our final conclusion is

# Fix(τ ∘ ϕ) = # Fix(ϕ) if ϕ ≠ id.

Of course, when ϕ = id, then τ ∘ ϕ is a pure translation, so # Fix(τ ∘ ϕ) is either ∞ or 0, according as whether τ = id. □

A.4f. Correspondence between morphisms and subgroups of J_{C}

Let (E, O) be an elliptic k-curve, where k is a perfect field. For each finite subgroup K ⊂ E(k̄) that is Gal(k̄/k)-stable, there is the non-constant separable k-isogeny

η_{K}: E [figure] \frac{E}{K} .

These are essentially the only non-constant separable isogenies with domain E, in the following sense. Let (E′, O′) be another elliptic k-curve. Every non-constant separable k-isogeny η: E [figure] E′ factors as the following composition of non-constant separable k-isogenies:

η: E [figure] \frac{E}{Ker(η)(k̄)} [figure: ∼] E′.

(See [Sil99, III.4.9, III.4.12].)

Of course, two different isogenies can have the same kernel (for example, consider id: E [figure] E and [−1]: E [figure]E), so one must be cautious in formulating the above in terms of a one-to-one correspondence. We see that non-constant separable k-isogenies E [figure]E′ are in one-to-one correspondence with pairs (K, E/K [figure: ∼] E′), where K is a finite Gal(k̄/k)-stable subgroup of E(k̄), and E/K [figure: ∼] E′ is a k-isomorphism.

We can eliminate “perfect” and “separable” from this correspondence by appealing to the language of schemes. Let (E, O) be an elliptic k-curve, where k is now an arbitrary field. For each finite subgroup k-scheme K ⊂ E, there is the non-constant k-isogeny

η_{K}: E [figure] \frac{E}{K} .

If (E′, O′) is another elliptic k-curve, then every non-constant k-isogeny η: E [figure] E′ factors as the following composition of non-constant k-isogenies:

η: E [figure] \frac{E}{Ker(η)} [figure: ∼] E′.

We can generalize this correspondence to curves of genus 1 as follows. Let C be a k-curve of genus 1. For each finite subgroup k-scheme K ⊂ J_{C}, there is the finite k-morphism

η_{K}: C [figure] \frac{C}{K} .

If C′ is another k-curve of genus 1, then every finite k-morphism η: C [figure] C′ factors as the following composition of finite k-morphisms:

η: C [figure] \frac{C}{Ker(η̂^{∗})} [figure: ∼] C′.

Here Ker(η̂^{∗}) is the finite subgroup k-scheme of J_{C} whose k̄-valued points are

Ker(η̂^{∗})(k̄) = { [P − Q] ∈ J_{C}(k̄) : P, Q ∈ C(k̄) and η(P) = η(Q) }.

Under the action of J_{C} on C, this subgroup acts simply transitively on each fiber of η; in particular, one can compute Ker(η̂^{∗})(k̄) by focusing on one fiber:

Ker(η̂^{∗})(k̄) = { [P − Q] ∈ J_{C}(k̄) : P, Q ∈ η^{−1}(R) }.

Since |η^{−1}(R)| = sep deg(η), the number of closed points in J_{C}[η] is sep deg(η). Multiplicities account for the inseparable degree contribution.

Just as, with a non-constant k-isogeny η: E [figure] E′ of elliptic k-curves there is a natural isomorphism Ker(η)(k̄) ≅ Aut(k̄(Ek̄)/η^{∗}k̄(E′ ; k̄)), with a finite k-morphism η: C [figure] C′ of k-curves of genus 1 there is a natural isomorphism

Ker(η̂^{∗})(k̄) [figure: ∼] Aut(k̄(Ck̄)/η^{∗}k̄(C′ ; k̄)).

Finally, the following statements are equivalent for a finite k-morphism η: C [figure] C′ of k-curves of genus 1 (cf. [Sil99, III.4.10]):

• η is separable;

• η is unramified;

• # Ker(η̂^{∗})(k̄) = deg η;

• The field extension k̄(Ck̄)/η^{∗}k̄(C′ ; k̄) is Galois. This leads to the following Galois correspondences when k is perfect:

• Finite subgroup k-schemes of J_{C} are in inclusion-preserving correspondence with Gal(k̄/k)- stable finite subgroups of Aut(k̄(Ck̄)/k̄).

• Either of the above sets is in inclusion-reversing correspondence with finite-index Gal(k̄/k)- stable subfields of k̄(Ck̄).

A.5. Consequences of Riemann–Hurwitz on curves of genus 1

We recall the Riemann–Hurwitz theorem for curves of arbitrary genus.

Theorem A.5.1 (Riemann–Hurwitz). For a finite separable morphism ϕ: X_{1} [figure] X_{2} between smooth projective curves,

2 genus(X_{1}) − 2 = deg(ϕ)(2 genus(X_{2}) − 2) + deg(R),

where R is the ramification divisor of ϕ on X_{1}, which, if the ramification is tame, looks like:

R = ∑_{P ∈X_{1}(k̄)} (e_{ϕ}(P) − 1)P.

Proof. See [Har77, IV.2.4]. □

Theorem A.5.2. If there exists a finite and purely inseparable morphism X_{1} [figure] X_{2} between smooth projective curves, then genus(X_{1}) = genus(X_{2}).

Proof. See [Har77, IV.2.5]. □

Corollary A.5.3. If ϕ: X_{1} [figure] X_{2} is a finite morphism between smooth projective curves, then

2 genus(X_{1}) − 2 ≤ sep deg(ϕ)(2 genus(X_{2}) − 2).  (A.1)

Proof. The morphism can be factored into a purely inseparable morphism followed by a separable morphism. Now apply the previous two theorems. □

Theorem A.5.4. The existence of a finite morphism X_{1} [figure] X_{2} between smooth projective curves implies genus(X_{1}) ≥ genus(X_{2}).

Proof. If genus(X_{2}) = 0, then there is nothing to prove. Otherwise, the right-hand side of (A.1) is non-negative, so we can eliminate sep deg(ϕ) to obtain 2 genus(X_{1}) − 2 ≥ 2 genus(X_{2}) − 2, which immediately gives the desired result. □

We can draw three consequences from Riemann–Hurwitz for curves of genus 1:

(1) The only finite morphisms with domain a smooth projective curve of genus 1 are: • finite maps between curves of genus 1, and

• finite maps to P^{1}, i.e., non-constant rational functions.

(2) Separable finite morphisms between smooth projective curves of genus 1 are unramified.

(3) More generally, the total ramification of a separable finite morphism from a curve of higher genus to a curve of genus 1 is independent of the degree of the morphism; however, at least with tame ramification, lower degrees force a higher number of branch points.

APPENDIX B

Maps to projective space: a coordinate-free approach

Let X be a scheme over a ring A, and let L be an invertible sheaf on X.

As described in [Har77, §II.7], if L is generated by global sections, then to each finite ordered collection B = (s_{0}, …, s_{n}) of global generators corresponds a unique morphism

ϕ_{B}: X [figure] P_{A}^{n}  (B.1)

with the properties ϕ_{B}^{∗}(O(1)) ≅ L and ϕ_{B}^{∗}(x_{i}) = s_{i}, where the x_{i} are the homogeneous coordinates on P_{A}^{n}. If the collection B is linearly independent, then the image of (B.1) is non-degenerate, meaning: it does not lie in a hyperplane of P_{A}^{n}.

By the universal property of the fibre product P_{X}^{n} := P_{A}^{n} ×_{A} X, each morphism (B.1) factors canonically as a morphism

ϕ_{B}: X [figure] P_{X}^{n}  (B.2)

followed by the canonical projection P_{X}^{n} [figure] P_{A}^{n}.

Our goal is to give a coordinate-free description of the map X [figure] P_{X}^{n} and of the composed map X [figure] P_{A}^{n}. To do so, we will require X and A to be noetherian, and we will furthermore require H^{0}(X, L) to be a free A-module of finite rank, or at least require a submodule of H^{0}(X, L) that generates L to have those properties. (By [Har77, II.5.19], these requirements are met if A is a field and X is projective over A.)

B.1. Background material

B.1a. Projective space bundles

Let X be a noetherian scheme. Associated to each locally free coherent sheaf E on X is the projective space bundle

π: P(E) [figure] X.  (B.3)

Let us recall its definition and properties (cf. [Har77, §II.7, p.162]). The sheaf

S := Sym(E) := ⨁_{d≥0} Sym^{d}(E)

is a quasi-coherent sheaf of graded O_{X}-algebras, where each homogeneous part S_{d} is coherent, S_{0} ≅ O_{X}, and S is locally generated by S_{1} = E. Thus we are in the situation denoted “(†)” in [Har77, §II.7, p.160], and by definition

P(E) := Proj(S).

In other words, for each affine open U ⊆ X, we have

π^{−1}(U) ≅ Proj(H^{0}(U, S)) = Proj(Sym(H^{0}(U, E))).

Furthermore, P(E) comes equipped with an invertible sheaf O(1) and there is a canonical surjective morphism π^{∗}(E) [figure] O(1), thus exhibiting O(1) as a “rank 1 quotient” of π^{∗}(E). Finally, if everywhere on X we have Rank(E) ≥ 2, then π_{∗}(O(d)) ≅ S_{d}; in particular, π_{∗}(O(1)) ≅ E. (See [Har77, II.7.11].)

Points correspond to rank 1 quotients. One way to think about P(E) is in terms of its T -valued points, which turn out to correspond to certain rank 1 sheaf quotients. Let g: T [figure] X be a T -valued point of X, which will remain fixed throughout this paragraph. If f: T [figure] P(E) is a T -valued point over X (i.e., π ∘ f = g), then f^{∗} carries the exact sequence

π^{∗}(E) [figure] O(1) [figure] 0

to the exact sequence

g^{∗}(E) [figure] f^{∗}(O(1)) [figure] 0.

[figure: P(E); X; π; T; g; f]

Note that f^{∗} is not only right-exact: it also preserves invertibility. Therefore, each T -valued point of P(E) over X gives rise to a rank 1 quotient of E on T, and this turns out to be a one-to-one correspondence (see [Har77, II.7.12]), which is easily verified to be functorial. In particular, when we exhibit a rank 1 quotient

g^{∗}(E) [figure] L [figure] 0,

the corresponding T -valued point f: T [figure] P(E) has the property

f^{∗}(O(1)) ≅ L.  (B.4)

Change of base. Projective space bundles behave well under change of base. Let h: X′ [figure] X be a morphism of noetherian schemes, and let E be a locally free coherent sheaf on X. We will now establish a morphism P(h^{∗}E) [figure] P(E) so that the diagram shown here commutes. One way to describe a morphism of schemes is to view the schemes as functors (assigning to each T the set of T -valued points) and then to describe a natural transformation of those functors. Thus, for each T -valued point f: T [figure] P(h^{∗}E), which immediately gives us the T -valued point π_{2} ∘ f of X′ and the T -valued point h ∘ π_{2} ∘ f of X, we must produce (in a

[figure: P(E); P(h^{∗}E); X′; π_{2}; X; h; π_{1}]

functorial fashion) a T -valued point of P(E) lying over h ∘ π_{2} ∘ f. By our correspondence between T -valued points and rank 1 quotients, f corresponds to

(π_{2} ∘ f)^{∗}(h^{∗}E) [figure] f^{∗}(O(1)) [figure] 0.

Since the left term is isomorphic to (h ∘ π_{2} ∘ f)^{∗}(E), the exhibited rank 1 corresponds to a T -valued point of P(E). It is furthermore not difficult (but left as an exercise for the reader) to establish that the diagram is cartesian:

P(h^{∗}E) = P(E) ×_{X} X′.  (B.5)

Coordinates. If E has finite rank everywhere on X, then the projective space bundle P(E) may be locally coordinatized as follows. Let U ⊆ X be an open set on which E is free and has constant finite rank, and set n so that the rank is n + 1. Let O_{U} denote the restriction of O_{X} to U. Let B = { s_{0}, …, s_{n} } be a basis of global sections establishing the isomorphism E|_{U} ≅ ⨁^{n} O_{U}. Observe that Sym(E|_{U}) is isomorphic, via the association s_{i} [figure] x_{i}, to the polynomial ring O_{U}[x_{0}, …, x_{n}]; thus, P(E|_{U}) ≅ P_{U}^{n}, and the structure map π: P(E|_{U}) [figure] U is then simply the canonical projection P_{U}^{n} [figure] U. Of course, if E is globally free, then a single such coordinatization works for the entire projective space bundle.

B.1b. Starting with a free A-module

Let X be a noetherian scheme over a noetherian ring A. If V is a free module of finite rank n + 1 over A, then E := V ⊗_{A} O_{X} is a locally free coherent sheaf of rank n + 1 on X. Here V ⊗_{A} O_{X} denotes the sheaf whose sections over an open set U ⊆ X are V ⊗_{A} H^{0}(U, O_{X}). (We view H^{0}(U, O_{X}) as an A-module as follows: via restriction it is certainly a module over H^{0}(X, O_{X}), and there is an obvious map A [figure] H^{0}(X, O_{X}) associating to each a ∈ A the corresponding constant function.) Similarly, V ⊗_{A} O_{A} is a locally free coherent sheaf of rank n + 1 on Spec(A). Observe

V ⊗_{A} O_{X} ≅ s^{∗}(V ⊗_{A} O_{A}),  (B.6)

where s: X [figure] Spec(A) is the structure map.

In fact, both V ⊗_{A}O_{A} and V ⊗_{A}O_{X} are globally free, whence they may be globally coordinatized. They can both be coordinatized simultaneously by choosing a basis B = { s_{0}, …, s_{n} } for V. Then the symbols s_{i} ⊗ 1 compose a basis of global generators both for V ⊗_{A} O_{A} and for V ⊗_{A} O_{X}.

In summary, we always have the commutative diagram

[figure: X; Spec(A); s; P(V ⊗_{A} O_{A}); P_{A}^{n},; ∼_{B}; P_{X}^{n}; P(V ⊗_{A} O_{X}); ∼_{B}; π]

where ∼_{B} indicates the isomorphism associated to a choice of basis B for V. By (B.6) and (B.5), describing an X-morphism X [figure] P(V ⊗_{A} O_{X}), i.e., a section of π, is equivalent to describing an A-morphism X [figure] P(V ⊗_{A} O_{A}).

B.1c. When the base is a field

When the base ring A is a field k, and K is a field extension of k, the following nice description of K-valued points of P(V) is useful. (Since Spec(k) has just a single point, the sheaf V ⊗_{k} O_{Spec(k)} is just the constant sheaf associated to V, so we might as well use V to denote both the vector space and the sheaf.)

Let X be a noetherian scheme over k. Let V be as before: a free k-module of finite rank n + 1. Fix a field extension K ⊇ k; in other words, choose a K-valued point g: Spec(K) [figure] Spec(k). We already know that a K-valued point of P(V) lying over g is nothing other than a rank 1 sheaf quotient of the sheaf g^{∗}V. But Spec(K) has a single point, so such a quotient amounts to a rank 1 vector space quotient of the vector space V ⊗_{k}K. This puts us squarely in the realm of linear algebra. By considering the kernel of a rank 1 quotient map, we see that rank 1 quotients are in one-to-one correspondence with hyperplanes. In summary, the K-valued points of P(V) are the hyperplanes in V ⊗_{k} K.

B.2. Coordinate-free version of X [figure] P_{X}^{n}

Let X be a noetherian scheme over a noetherian ring A, and let L be an invertible sheaf on X that is generated by global sections. Viewing H^{0}(X, L) as an A-module, let V ⊆ H^{0}(X, L) be a submodule with the following properties:

• V generates L;

• V is a free A-module of finite rank. We claim that, under these assumptions, the projective space bundle

π: P(V ⊗_{A} O_{X}) [figure] X.

has a canonical section

ϕ_{V} : X [figure] P(V ⊗_{A} O_{X});  (B.7)

furthermore, there is a canonical isomorphism ϕ_{V}^{∗} (O(1)) ≅ L, and if we choose a basis B = { s_{0}, …, s_{n} } for V, whence P(V ⊗_{A} O_{X}) ≅ P_{X}^{n}, then we obtain the original map (B.2) described at the outset.

Before establishing the claim, let us remark on consequences. If we compose (B.7) with P(V ⊗_{A} O_{X}) [figure] P(V ⊗_{A} O_{A}), we end up with

ϕ_{V} : X [figure] P(V ⊗_{A} O_{A}).  (B.8)

Under the basis B, (B.8) recovers the original map (B.1). Since (B.1) pulls back an A-basis of H^{0}(P_{A}^{n}, O(1)) to an A-basis of H^{0}(X, L), pullback must give a canonical isomorphism of A-modules

H^{0}(P(V ⊗_{A} O_{A}), O(1)) ≅ H^{0}(X, L).  (B.9)

In particular, the image of (B.8) is non-degenerate, meaning: it does not lie in a hyperplane. That is, no nonzero global section of O(1) vanishes identically on X.

Example. When A is a field k, X is a curve, and ϕ_{V} is the map associated to V = H^{0}(C, O(D)), where D is a divisor on X, (B.9) says: equations for hyperplanes in P(V) are in one-to-one correspondence with { f ∈ k(X) : (f) + D ≥ 0 }. This leads to a familiar fact: if we remove 0 from each vector space in (B.9) and then work modulo k^{×}, classes on the left correspond to hyperplanes, while classes on the right correspond to hyperplane sections: effective divisors linearly equivalent to D. In short, (B.9) establishes a one-to-one correspondence between hyperplanes and hyperplane sections. When ϕ_{V} is an embedding, the correspondence is simply H [figure] H ∩X; in general, one must instead choose an equation ℓ defining H, pull back ℓ to obtain a rational function ϕ_{V}^{∗} ℓ on X, and finally take the divisor of ϕ_{V}^{∗} ℓ.

Proof of claim. We start by describing (B.7). Given a T -valued point g: T [figure] X, we must produce (in a functorial fashion) a T -valued point T [figure] P(V ⊗_{A} O_{X}). We saw earlier that this is the same as producing a rank 1 quotient of g^{∗}(V ⊗_{A} O_{X}). To do this, we simply apply g^{∗} to the given rank 1 quotient V ⊗_{A} O_{X} [figure] L [figure] 0 on X. (The map V ⊗_{A} O_{X} [figure] L is described as follows. To each global section s ∈ V corresponds the morphism O_{X} [figure] L defined by f [figure] f · s. Given a pair (s, f), where s ∈ V and f ∈ H^{0}(U, O_{X}), we “evaluate” s on f to obtain f · s ∈ H^{0}(U, L). This pairing is bilinear and thus gives the map in question. It is surjective precisely because V generates L.)

Having described (B.7), we now establish ϕ_{V}^{∗} (O(1)) ≅ L. To do so, we again appeal to the description of ϕ_{V} as a natural transformation of scheme functors. For each T -valued point g: T [figure] X, we must establish an isomorphism of sheaves

g^{∗}(L) ≅ ϕ_{V} (g)^{∗}(O(1)).

The map ϕ_{V} (g) corresponds to the rank 1 quotient g^{∗}(V ⊗_{A} O_{X}) [figure] g^{∗}(L) [figure] 0, and thus, by (B.4), the pullback of O(1) is indeed isomorphic to g^{∗}(L).

Finally, to see that a choice of basis B = { s_{0}, …, s_{n} } for V leads to our original map (B.2), it remains to verify, by the characterizing properties stated after (B.1), that the sections x_{i} in O(1) on P_{A}^{n} pull back to the sections s_{i} on X. We are looking at the composition

X [figure: ϕ_{V}] Proj(Sym(V ⊗_{A} O_{X})) [figure: B] Proj(O_{X}[x_{0}, …, x_{n}]) = P_{X}^{n}.

Under pullback, x_{i} on P_{X}^{n} corresponds to s_{i} ⊗1 on Proj(Sym(V ⊗_{A} O_{X})), which in turn corresponds to s_{i} on X. □

When A is a field k, (B.8) has an elegant description in terms of K-valued points, where K is a field extension of k. We already know that a fixed K-valued point g: Spec(K) [figure] X goes to the rank 1 quotient

g^{∗}(V ⊗_{k} O_{X}) [figure] g^{∗}L [figure] 0,

which gives a point on P(V ⊗_{k} O_{X}). Under P(V ⊗_{k} O_{X}) [figure] P(V), that point goes to the rank 1 quotient

(s ∘ g)^{∗}(V) [figure] g^{∗}L [figure] 0,

which is a point on P(V), where s is the structure map X [figure] Spec(k). Since Spec(k) has a single point, the data of (s ∘ g)^{∗}(V) is simply the K-vector space V ⊗_{k} K. (Cf. B.1c.) The kernel of

V ⊗_{k} K [figure] g^{∗}L

is the hyperplane of all elements in V ⊗_{k} K that map to 0 in the fiber of L above the point of X hit by g. In short, ϕ_{V} carries the K-valued point g to the hyperplane of sections of L that vanish at g. Therefore, it is not uncommon to see (B.8) described solely in the following terms:

\begin{matrix}ϕ_{V} : X(K) & [figure] P(V)(K), \\ P & [figure] \{ s ∈ V ⊗_{k} K : s_{P} = 0 \}. \end{matrix}

B.3. Mapping properties

Perhaps surprising at first, it is in general difficult, as we’ll see below, to extend morphisms between schemes, or between sheaves, to morphisms between projective space bundles.

B.3a. Morphism of schemes

Let f: X′ [figure] X be a morphism of noetherian schemes over a noetherian ring A. If L is an invertible sheaf on X that is generated by global sections and such that H^{0}(X, L) is a free A-module of finite rank n+1, then f^{∗}L is an invertible sheaf on X′ that is generated by global sections, but H^{0}(X′, f^{∗}L) could have much larger rank. It would be nice if there existed maps f_{∗} giving a commutative diagram:

[figure: X′; P(H^{0}(X′, f^{∗}L) ⊗_{A} O_{X′}); ϕ_{f^{∗}L}; P(H^{0}(X′, f^{∗}L) ⊗_{A} O_{A}); P(H^{0}(X, L) ⊗_{A} O_{A}); f_{∗}; P(H^{0}(X, L) ⊗_{A} O_{X}); X; ϕ_{L}; f; f_{∗}]

But there do not exist non-constant morphisms from a higher-dimensional projective space to a lower-dimensional one (see [Har77, Ex. II.7.3a]), whence a necessary condition for the existence of f_{∗} is that H^{0}(X′, f^{∗}L) be a free A-module of rank n+1. By applying the correspondence between T -valued points and rank 1 quotients, one can check that existence of the f_{∗} follows if the map

f^{∗}(H^{0}(X, L) ⊗_{A} O_{X}) [figure] H^{0}(X′, f^{∗}L) ⊗_{A} O_{X′}

of sheaves on X′ is surjective. This gives a sufficient condition.

The one easy thing we can say is this: if f is an isomorphism of schemes, then we do obtain the maps f_{∗}, and they will then also themselves be isomorphisms.

B.3b. Morphism of sheaves

Here it turns out, again, that we can’t make a useful general statement. Let X be a noetherian scheme over a noetherian ring A. Let f: L [figure] L′ be a morphism of invertible sheaves on X, where both sheaves are generated by their global sections, and where both H^{0}(X, L) and H^{0}(X, L′) are free A-modules of finite rank (not necessarily the same rank). It would be nice if there existed a map f^{∗} giving a commutative diagram:

[figure: X; P(H^{0}(X, L) ⊗_{A} O_{A}); P(H^{0}(X, L′) ⊗_{A} O_{A}); f^{∗}; X]

To be able to define such a map f^{∗}, we would need the induced map H^{0}(X, L) [figure]H^{0}(X, L′) to be surjective; otherwise we cannot guarantee that rank 1 quotients go to rank 1 quotients. But then f itself would be surjective, so by [Har77, Ex. II.7.1], f was an isomorphism to begin with. In summary, all we have been able to say here is the following: if f: L [figure: ∼] L′ is an isomorphism of sheaves on X, then there is an induced commutative diagram as shown above, where f^{∗} is an isomorphism.

B.3c. Different subspaces of H^{0}(X, L)

Let X be a noetherian scheme over a noetherian ring A, and let L be an invertible sheaf on X. If we have submodules V ⊆ V′ ⊆ H^{0}(X, L), where V generates L (whence also V′ generates L), and both V and V′ are free and of finite rank (generally not the same rank) as modules over A, then the inclusion map V [figure] V′ leads to a projection map

[figure: P(V′ ⊗_{A} O_{A}); P(V ⊗_{A} O_{A}).]

Note that the projection is not defined everywhere; however, it is defined at all points in the image of ϕ_{V′}. If B is a basis for V and B′ a basis for V′ such that B ⊆ B′, and we use these bases to coordinatize the projective space bundles, then the projection displayed above becomes a usual projection between projective spaces in which certain homogeneous coordinates are simply dropped.

[figure: X; P(V′); ϕ_{V ′}; P_{A}^{Rank(V′)}; ∼_{B′}; P_{A}^{Rank(V)}; P(V); ∼_{B}; X; ϕ_{V}]

B.4. Very ample invertible sheaves

Let X be a noetherian scheme over a noetherian ring A. To each invertible sheaf L that is generated by global sections and such that H^{0}(X, L) is a free A-module of finite rank n + 1, we have associated the map

ϕ: X [figure] P(H^{0}(X, L) ⊗_{A} O_{A}).

We say that L is very ample relative to A when ϕ is an immersion. (Note that the map to P(H^{0}(X, L) ⊗_{A} O_{X}) is always an immersion, so L is always very ample relative to X itself.)

If we forget about L and instead start with an immersion ϕ: X [figure] P_{A}^{n}, then ϕ^{∗}(O(1)) will be an invertible sheaf on X that is very ample relative to A. (After all, the map to projective space associated with ϕ^{∗}(O(1)) is easily seen to be ϕ itself.)

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