Integrated MSc-PhD Program


KSM4E09: Modular Forms

Topics covered include the modular group and congruence subgroups, the definition of modular forms and their first properties, Eisenstein series, theta series, valence formulas, Hecke operators, Atkin-Lehner-Li theory, L-functions, modular curves, and modularity. The final lectures will provide a global outlook on the use of modular forms (and their associated Galois representations) in solving Diophantine equations, particularly Fermat’s Last Theorem.

Suggested texts:

          1. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, Springer-Verlag, 2005.
          2. Toshitsune Miyake, Modular Forms.
          3. J.S. Milne, Modular Functions and Modular Forms, online course notes.
          4. J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.