M4P58: Modular Forms
Tue 2-3, Wed 10-11, Thu 10-11 in 642 Huxley
Dr. David Helm
Office Hours: Mon 3-5
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.
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.
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
Problem sets will be posted here every two weeks. They will not be assessed work.
Example Sheet 1 (Due Thursday 5 November)
Example Sheet 2 (Due Thursday 19 November)
Example Sheet 3 (Due Thursday 3 December)
Example Sheet 4 (Due Thursday 17 December)