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MA6H9 Modular Forms

Lecturer: Samir Siksek

Term(s): Term 2

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

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

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)

Additional Resources