MODULAR FORMS

I noticed that my course description for the Part III Modular Forms course didn’t seem to make it into the list which was sent out to new Part III students. Apologies for this. In an attempt to make amends, here it is.

Kevin Buzzard

Course description:

Modular forms were initially studied by such people as Siegel and Ramanujan in the early 20th century, but then went out of fashion until the 1960s, when very profound connections were discovered between modular forms and several other branches of number theory. Initially modular forms were thought of as analytic objects (modular forms are certain differentiable functions on the complex upper half plane) but it was realised in the 1960s that their Fourier expansions encoded a lot of algebraic data, related to, for example, Galois groups, elliptic curves, and L-functions.

Further secrets of modular forms were discovered in the late 1980s, when Frey and Ribet found a way to link modular forms to the solution of Diophantine equations. A few years later, by proving theorems about modular forms, Wiles and Taylor were able to deduce Fermat’s Last Theorem. Other Diophantine equations have also been solved using these ideas.

An introductory lecture course on modular forms can only hope to scratch the surface of this huge subject. In particular, I shall probably not get to the proof of Fermat’s Last Theorem (or indeed, anywhere near it). What I shall do is talk about the basics of the theory. I shall start with the theory of modular curves and the analytic definition of a modular form. Next I will spend some time proving some basic results about spaces of modular forms, and giving lots of examples. I shall cover Hecke operators and their effect on the Fourier expansion of a modular form. We should then be in a position to say some basic things about L-functions. This will probably take up most of the course, but I shall hopefully have a lecture left at the end to talk about the subtle links between modular forms and Diophantine equations.

Prerequisites: some undergraduate complex analysis, basic group theory, and linear algebra would be very helpful.

Appropriate books:

The last chapter of J.-P. Serre, ”A Course in Arithmetic” is a very well-written introduction to the theory of modular forms, and is in fact where I learnt the basics.

N. Koblitz ”Introduction to elliptic curves and modular forms” is very much what its title suggests.

T. Miyake, ”Modular Forms” contains almost everything that I shall cover. Unfortunately (as I recall) I don’t think the UL has it.

G. Shimura, ”An introduction to the Arithmetic Theory of Automorphic Forms” contains far more than I can hope to cover, and all in much greater generality.