# On the projective geometry of curves of genus one, and an algorithm for the jacobian of such a curve

On the projective geometry of curves of genus one, and an algorithm for the jacobian of such a curve.

Ph.D. Dissertation, 2004.

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.

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Last change: 16 July 2004.

## Related work:

- The dissertation was a continuation of the work in our paper on jacobians.