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]