M4P58: Modular Forms

Tue 12-1, Thu 10-11 in 642 Huxley, Fri 11-12 in 658 Huxley
Dr. David Helm
672 Huxley
Office Hours: Mon 4-5, Tue 2-3 (or by appointment)

Course description

This course concerns the theory of modular forms, which are holomorphic functions on the complex upper half plane that exhibit a very high degree of symmetry. Such functions have surprising applications outside of analysis; for example, their power series expansions often encode useful arithmetic information such as the number of solutions of certain diophantine equations modulo various primes. In this course we will establish the foundations of the theory of modular forms, and related functions such as elliptic functions, and illustrate some of the applications to arithmetic.

Suggested References

Serre's A Course in Arithmetic, Chapter VII, is a classic reference for the theory of modular forms of level one, and we will follow it fairly closely for the first six weeks of the course. The first chapter of Silverman's Advanced topics in the the arithmetic of elliptic curves is another good reference, and covers some material, such as the theory of elliptic functions, that Serre omits.
For the theory of modular forms of higher level, the references are less standard. One place to look is Apostol's Modular Functions and Dirichlet Series in Number Theory, but this does not discuss Hecke operators at higher level. Diamond and Shurman's A First Course in Modular Forms certainly covers everything we will cover (and much more!), but freely uses much more machinery level than the course will assume. There are also Milne's modular forms lecture notes which make more use of geometry- and in particular the theory of Riemann surfaces- than we will, but which might be useful nonetheless.


Problem sets will be posted here every two weeks. They will not be assessed work. There will be problem session on the days they are due.
This year the due dates (and problem sessions) will be: Tuesday 1 November, Tuesday 15 November, Tuesday 29 November, and Tuesday 13 December.
Example Sheet 1 (Due Tuesday 1 November) Solutions
Example Sheet 2 (Due Tuesday 15 November) Solutions
Example Sheet 3 (Due Tuesday 29 November) Solutions
Example Sheet 4 (Due Tuesday 13 December) Solutions