Elliptic Curves over Function Fields and an Analog of
the Gross-Zagier Theorem
- Review of elliptic curves over function fields
- Automorphic forms and analytic modularity
- Drinfeld modular curves and geometric modularity
- Overview of the Gross-Zagier computation
I will assume that the audience is familiar with the basic facts about
elliptic curves (over Q), say at the level of Silverman's
book. I'll also assume the basics of curves over finite fields and
their functions fields, such as adeles, vector bundles/locally free
sheaves, and a tiny bit of cohomology (H^1 of a line bundle).
Although my discussions of automorphic forms and Drinfeld modular
curves will start from the beginning, some knowledge of their classical
counterparts should be quite helpful.
For those who want to prepare a little, here are suggestions keyed to
- The main point of the lecture will be to explain results of Tate
(Bourbaki 306) and refinements by Milne (Annals 102) on the conjecture
of Birch and Swinnerton-Dyer. The best preparation would be to
convince oneself that the main results and conjectures about elliptic
curves over number fields (Mordell-Weil, L-functions, heights, sha,
BSD) also make sense over function fields. At the end of the lecture,
a willingness to believe in a good (l-adic) cohomology theory will be
- Chapter I of Weil's "Dirichlet Series and Automorphic Forms"
(Lecture Notes 189) explains the connection between classical modular
forms (functions on the upper half plane) and functions on adelic
matrices. The function field version of these matrices will be my
starting point. Chapter III of the same book gives a painless
introduction to basics like Fourier expansions. (This whole volume,
although out of fashion, is well worth reading, and not too
- There is a vast literature on Drinfeld modules and Drinfeld
modular curves, but a quick reference that is more than enough to get
up and running is the first three chapters of Deligne and
Husemöller's survey in Contemporary Math 67.
- Chapter I
of Gross-Zagier (Inventiones 84) and the introductions to the
following chapters are still the best overview of their work. The
function field version is simply slavish imitation, with a few
complications caused by working with a general base field.