## M4P58: Modular Forms

**Tue 12-1, Thu 10-11 in 642 Huxley, Fri 11-12 in 658 Huxley**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

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.
### Problems

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