# 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.

### Lecture notes

Lecture 1 (11 Oct): Motivation. Locally profinite groups. Smooth and admissible representations. The smooth dual. Induction from subgroups. The modulus character.

Lecture 2 (18 Oct): Duality for induced representations. GL(2): standard subgroups and Bruhat / Iwasawa / Cartan decompositions. Principal series representations $I(\chi, \psi)$ and their decomposition into irreducibles (statement and sketch of proof).

Lecture 3 (25 Oct): Hecke algebras. The spherical Hecke algebra for GL(2). Unramified principal series. The Iwahori-Hecke algebra. Statement of Casselman's new vectors theorem and uniqueness of Whittaker functionals.

Lecture 4 (1 Nov): The Kirillov model; proof of new vectors theorem. Adeles and ideles. Strong approximation for SL(2). Modular curves as adelic double quotients.

**NB: No lecture 8th Nov.**

Lecture 5 (15th Nov): More on adelic double quotients. Modular forms as functions on adele groups; dictionary between classical and adelic Hecke operators. Hilbert modular forms (brief sketch). Restricted tensor products and Flath's tensor product theorem. Global Kirillov models.

## Problem sheets

Sheet 1 (18 Oct) (covers lectures 1–2) — Solutions

Sheet 2 (15th Nov) (covers lectures 3-5). Deadline **7th December.**

### 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. Existence and uniqueness of Kirillov models (without proof). Casselman's local new vector theorem.**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]