M4P58: Modular Forms

Fri 11-1 in 642 Huxley, Mon 5-6 in 658 Huxley
Dr. David Helm
672 Huxley
Office Hours: Fri 2-4 (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.

Lecture notes

Robert Kurinczuk has kindly let me use a copy of his lecture notes for his course two years ago, which will be very close in content to the current one. They can be found here.

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 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.

Assessed Coursework

There will be two assessed courseworks assigned, each worth 5 percent of your total marks. They should be handed in at the Undergraduate Office by 4pm on the day they are due. The first will be due on Monday, 4 November, and the second will be due on Monday, 2 December. The courseworks will be posted here at least two weeks in advance of the due dates.
Assessed Problems 1 (Due Monday 4 November) Solutions
Assessed Problems 2 (Due Monday 2 December)

Discussion Problems

In addition to the assessed coursework, I will be assigning additional coursework in the form of discussion problems. These are more challenging problems that will be discussed in problem sessions on the date they are due; they will not be assessed work. Since these questions are more difficult than typical assessment or exam questions, you are encouraged to work on them in groups and share ideas and approaches (and of course I am happy to give hints in office hours!) The discussion problems will be due on the following Mondays: 28 October, 18 November, and 9 December and posted here at least two weeks before those dates.
Discussion Problems 1 (Due Monday 28 October) Solutions
Discussion Problems 2 (Due Monday 18 November) Solutions
Discussion Problems 3 (Due Monday 9 December) Solutions