MA4H9 Modular Forms
Lecturer: Samir Siksek
Term(s): Term 2
Status for Mathematics students: List C
Commitment: 30 lectures, plus a willingness to work hard at the homework
Assessment: 100% 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)