# Modular Forms and Representations of GL(2) (TCC 2018)

Lectures are 14.00–16.00 on Thursdays, beginning on 11th October, in whichever room your department uses for TCC courses. For Warwick students this is Zeeman Building B0.06.

### Syllabus (provisional)

**Motivation.**Modular forms and Hecke operators.**Representations of locally profinite groups**. Smooth and admissible representations. The smooth dual. Induction from subgroups. The modulus character and normalised induction.**Hecke algebras.**Definition of the Hecke algebra. Dictionary between representations and Hecke modules.**GL(2) over a p-adic field**. Cartan and Iwasawa decompositions. The principal series. The spherical Hecke algebra and the Satake transform. Classification of irreducible spherical representations. Statement of Casselman's local new vector theorem (without proof).**GL(2) over the adeles**. Definition of the adele ring and its topology. Flath's tensor product theorem. Strong approximation, geometry of adelic double quotients. Modular forms as functions on adele groups. Dictionary between classical and adelic Hecke operators. The global multiplicity one theorem.

### Prerequisites

Students will need to be familiar with the basic concepts of representation theory of finite groups, and of the arithmetic of p-adic fields. Knowledge of modular forms is useful for motivation, although it is not technically required except for the last section.

### References

- Bump,
*Automorphic forms and representations* - Gelbart,
*Automorphic forms on adele groups*

- Jacquet and Langlands,
*Automorphic forms on GL(2),*Springer Lecture Notes #114 [for the very ambitious!] - Casselman,
*Introduction to admissible representations of p-adic groups*

- Bushnell and Henniart,
*The local Langlands conjecture for GL(2)*[early chapters only]