# MA6H9 Modular Forms

**Lecturer: Samir Siksek**

**Term(s):** Term 2

**Commitment:** 30 lectures, plus a willingness to work hard at the homework.

**Assessment:** 100% by oral examination

**Formal registration prerequisites: **None

**Assumed knowledge:**

**Useful knowledge: **Core first and second year modules

**Synergies: **This module complements the following:

**Leads To: **Ph.D. studies in number theory and algebraic geometry

**Content**: The course's core topics are the following:

- 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

**Books**:

F. Diamond and J. Shurman, *A First Course in Modular Forms*, Graduate Texts in Mathematics 228, Springer-Verlag, 2005. (Covers everything in the course and a great deal more, with an emphasis on introducing the concepts that occur in Wiles' work)

J.-P. Serre, *A Course in Arithmetic*, Graduate Texts in Mathematics 7, Springer-Verlag, 1973. (Chapter VII is a short but beautifully written account of the first part of the course which is good introductory reading)

W. Stein, *Modular Forms, A Computational Approach*, Graduate Studies in Mathematics, American Mathematical Society, 2007. (Emphasis on computations using the open source software package Sage)