# MA4H9 Content

**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.)