## M4P58: Modular Forms

**Fri 11-1 in 642 Huxley, Mon 5-6 in 658 Huxley**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

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