Introductory description

Modular forms are holomorphic functions on the complex upper half plane that enjoy certain additional symmetries. Whilst the definition is analytic they have many applications within number theory and algebra. In this module we study spaces of modular forms and their connections to L-functions, quadratic forms and elliptic curves.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The modular group and the upper half-plane.
Modular forms of level 1 and the valence formula.
Eisenstein series, Ramanujan's Delta function.
Congruence subgroups and fundamental domains. Modular forms of higher level.
Hecke operators.
The Petersson scalar product. Old and new forms.
Statement of multiplicity one theorems.
The L-function of a modular form.
Modular symbols

Learning outcomes

By the end of the module, students should be able to:

• be able to perform simple computations involving modular forms and their Hecke operators
• be able to derive and use dimension formulae for spaces of modular forms
• appreciate the link between modular forms and elliptic curves via L-functions

Subject specific skills

Use methods of complex analysis to prove algebraic and number theoretic statements
Use linear algebra to prove identities between modular forms
Use the Poisson summation formula to derive transformation laws for theta series

Transferable skills

Ability to use Cauchy's residue theorem. Ability to translate between the analytic and algebraic worlds. Ability to solve problems.