Math7280 : Overview

The following brief description is exactly as in the course descriptions web page, but more details are given below on topics, prerequisites, presentation and assessment. See also the references and links pages.

Brief description

Text: Introduction to the arithmetic theory of automorphic forms, G. Shimura. (Strongly recommended, but not required; used as reference material; we will cover roughly the first 4 chapters; some topics not covered in the text will be presented as handouts.)

This course is primarily an introduction to the theory of modular forms, with applications to elliptic curves. A modular form is an analytic function on the upper half complex plane having many symmetries, and an expansion of the form f(z) = a₀ + a₁ q + a₂ q² + a₃ q³ + .... where q=exp(2pi iz). An elliptic curve is (usually) given by an equation y² = x³ + ax + b (and has only an historical connection to ellipses). The modularity of elliptic curves is the surprising fact that for every elliptic curve (with a, b rationals), there is a corresponding modular form such that the number of points on the curve, mod p, for all but finitely many primes p, is p minus the coefficient a_p of the modular form.

Some of the topics we will cover are: Congruence subgroups, fundamental domains for subgroups of PSL₂(Z), Hecke operators, the group law of an elliptic curves, p-adic numbers, and Galois representations associated with elliptic curves. A goal is to be able to sketch an outline of the ingredients that go into how the modularity of elliptic curves is used to prove Fermat's last theorem. We will also discuss computational problems, such as how to compute Hecke eigen modular forms, and fundamental domains. These will be illustrated with the Magma computer algebra package.

Topics

In more detail, the topics we will cover are as follows:

&bull Definitions of modular forms, fundamental domains, modular curves, Hecke operators

&bull Elliptic curves, and their group law and zeta functions and L-series

&bull Modularity of elliptic curves

This is the relationship between the coefficients of modular forms and the number of points on elliptic curves. (an example is given on the index page.) Note, this is a very deep theorem (proved by Wiles, Taylor, Breuil, Conrad and Diamond, and previously known as the Taniyama-Shimura-Weil, or modularity conjecture), so we will not prove this, except perhaps in special cases, but we will discuss what it means, and some applications, such as the proof of Fermat's last theorem.

&bull Congruences for coefficients of modular forms

An example is the mod 691 congruence on this index page. Again, to prove this kind of relationship in general will be beyond this course, since we need to prove the existence of Galois representations corresponding to certain modular forms. However, given such a correspondence, we will be able to show how these kinds of congruence properties arise.

&bull l-adic numbers, infinite Galois groups, l-adic Galois representations, and modular Galois representations

Both of the above phenomena (modularity of elliptic curves, and congruences for the coefficients of certain modular forms) are proved by using l-adic Galois representations. In this course we will define l-adic numbers, infinite Galois groups, and l-adic Galois representations, only assuming knowledge of finite fields, and Galois groups of finite extensions of Q. We will only prove modularity of Galois representations in special cases, but we will describe what it means in general.

Prerequisites

A few topics you should be familiar and comfortable with, and know basic definitions and examples. You do not need to have a huge amount of background on these. If you are uncertain if you have enough background, come and ask me (office 210). I can suggest further background reading if necessary.

Undergraduate complex analysis, e.g., meromorphic and analytic functions
Finite fields
Projective space
Groups and group actions
Galois groups, especially for finite extensions of Q
Riemann surfaces

Form of presentation

Lecturing is done at the black board, by the professor, except perhaps a few student presentations right at the end. To begin with we will follow Shimura's book fairly closely. When we come to elliptic curves and p-adic numbers, and l-adic Galois representations, we will cover these in more detail, assuming no knowledge of either. Handouts will supplement the book here. For the first half of the course we will prove pretty much everything we cover. For the second half, in order to be able to discuss some more interesting examples, we will sometimes quote and explain the meaning and applications of results, rather than giving proofs of everything. In some lectures I will use a computer, projecting onto a screen, to demonstrate some examples computed with magma.

Assessment

Assessment of the attendee's understanding of the material presented, will be through homework problems, and probably also a final short project, which would involve preparing a 30 to 60 minute talk on a topic on modular forms and/or elliptic curves which was not lectured on, but which would be closely related to the material that will be covered. Exact details to be decided.

Links

For Spring 2004, by H. Verrill, 27 Oct 03