## M4P58: Modular Forms

**Mon 12-1, Tue 10-11, Fri 3-4 in 658 Huxley**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

Office Hours: Tue 1-3

### 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. 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.
Diamond and Shurman's *A First Course in Modular Forms* certainly
covers everything we will cover (and much more!), but is at a somewhat higher level than the course will be taught at.
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.

Example Sheet 1 (Due Friday 7 November)
Solutions

Example Sheet 2 (Due Friday 21 November)
Solutions

Example Sheet 3 (Due Friday 5 December)
Solutions

Example Sheet 4 (Due Friday 19 December)
Solutions (partial)