# MA4H9 Modular Forms

**Lecturer:** Guhanvenkat Harikumar

**Term(s):** Term 1

**Status for Mathematics students:** List C

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

**Assessment:** 85% by 3-hour examination, 15% assessed work.

**Prerequisites:** MA231 Vector Analysis. Additionally, MA3B8 Complex Analysis is highly recommended although not strictly required. Please talk to me if you have not yet taken MA3B8 and still want to take this course.

**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. 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.)