# 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.

Lecture 5 (15 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.

Lecture 6 (22 Nov): Proof of global multiplicity one + strong multiplicity one theorems. Consequences for classical modular forms. Twisting automorphic representations. Fourier–Whittaker expansions for Hilbert modular forms. Eisenstein series (reminders from classical theory).

Lecture 7 (29 Nov): Eisenstein series (adelic viewpoint). Rational structures on modular forms spaces, canonical $\mathbb{Q}$-models of modular curves.

Lecture 8 (6 Dec): L-functions of modular forms. The Rankin–Selberg method (after Jacquet).

## Problem sheets

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

Sheet 2 (15th Nov) (covers lectures 3-5) — Solutions

Sheet 3 (6th Dec) (covers lectures 6-8) — Solutions

### Prerequisites

For the first half of the course, students will just need to be familiar with the basic concepts of representation theory of finite groups, and of the arithmetic of p-adic fields. Familiarity with the definitions and basic properties of modular forms is needed from Lecture 5 onwards.

### 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]