Math7280 : Overview
The following brief description is exactly as in the
descriptions web page, but more details are given below on
topics, prerequisites, presentation and assessment.
See also the
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
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 ap 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.
In more detail,
the topics we will cover are as follows:
Definitions of modular forms, fundamental domains,
modular curves, Hecke operators
Elliptic curves, and their group law and zeta functions
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
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.
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.
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.
A few topics you should be familiar and comfortable
with, and know basic definitions
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 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.
For Spring 2004, by H. Verrill, 27 Oct 03