This seminar is aimed primarily at the graduate students in the number theory group. The seminar this term will be devoted to the topic of the Sato-Tate conjectures, using as a basic reference Carayol's Bourbaki seminar lecture, but filling in a lot of background along the way, with special emphasis on the analytic aspects of the argument. We will strive to present the more technical parts of the proof as a black box, so that the prerequisites required to follow the seminar will be minimised. In particular, while the knowledge acquired from last year's seminar on the Shimura-Taniyama conjecture and modularity lifting theorems will certainly be helpful, it will neither be assumed or required.

The real goal of the seminar will be to explain how, assuming a fairly general Wiles-style ``modularity lifting theorem", one can deduce the Sato-Tate conjecture.

Kushal Banerjee

Joachim de Ronde

Michele Fornea

Ananyo Kazi

David Lilienfeldt

Simone Maletto

Kunjakanan Nath

Isabella Negrini

Alice Pozzi

James Rickards

Andrei Shubin

Nicolas Simard

Peter Zenz.

In addition the following faculty and post-docs will be attending:

Henri Darmon

Chantal David

Adrian Iovita

Dimitris Koukoulopoulos

Maksym Radziwill

Giovanni Rosso

Jan Vonk

We will be using the Bourbaki seminar article of Carayol devoted to the proof of the Sato-Tate conjecture as a

Serre's landmark book Abelian l-adic representations and elliptic curves , W.A. Benjamin, 1968 - based on a course he taught at McGill in 1967, exactly 50 years ago! - contains valuable background on l-adic representations and a discussion of the relation between equidistribution of frobenius eigenvalues and meromorphic continuation and zero-free regions for symmetric power L-functions.

- September 28.
**Dimitris Koukoulopoulos**. The first lecture will begin with a statement of the Sato-Tate conjecture and an explanation of why the existence of a zero-free region for the L-functions attached to symmetric power representations implies the Sato-Tate conjecture. Dimitris will give a pretentious introduction to the subject. - October 5.
**Peter Zenz, Peter Zenz**. A less pretentious explanation of why the existence of a zero-free region for the L-functions attached to symmetric power representations implies the Sato-Tate conjecture, following the treatment given in Serre's book. - October 12.
**Peter Zenz, Kunkannn Nath**. A less pretentious explanation of why the existence of a zero-free region for the L-functions attached to symmetric power representations implies the Sato-Tate conjecture, following the treatment given in Serre's book. - October 19.
**Isabella Negrini**. An overview of the Sato-Tate conjecture, following the expository article by Drew Sutherland. - October 26.
**James Rickards**. An overview of the Sato-Tate conjecture, following the expository article by Drew Sutherland, cont'd. - November 2.
**Simone Maletto**. Galois representations, compatible system and their L-functions (d'après Taylor) following Taylor's ICM lecture notes (or the extended version in Journal de Toulouse, both of which are available on Taylor's web page.) Local Galois representations; p-adic representations of local fields of residue characteristic l, with l not equal to p. Weil-Deligne representations. - November 9.
**Simone Maletto**, cont'd. Galois representations, compatible system and their L-functions (d'après Taylor) following Taylor's ICM lecture notes (or the extended version in Journal de Toulouse, both of which are available on Taylor's web page.) Local Galois representations; p-adic representations of local fields of residue characteristic l, with l not equal to p. Weil-Deligne representations. - November 16.
**Ananyo Kazi**. Galois representations, compatible system and their L-functions (d'après Taylor) following Taylor's ICM lecture notes (or the extended version in Journal de Toulouse, both of which are available on Taylor's web page.) Local Galois representations; p-adic representations of local fields of residue characteristic p. The rudiments of p-adic Hodge theory. - November 23.
**Alice Pozzi**Galois representations, compatible system and their L-functions (d'après Taylor) following Taylor's ICM lecture notes (or the extended version in Journal de Toulouse, both of which are available on Taylor's web page.) Cont'd. - November 30.
**Joachim de Ronde.**Galois representations, compatible system and their L-functions (d'après Taylor) following Taylor's ICM lecture notes (or the extended version in Journal de Toulouse, both of which are available on Taylor's web page.) Cont'd. - December 7.
**David Lillienfeldt**Automorphic forms for GL(n) and their L-functions (d'après Taylor) - December 14.
**Kushal Banerjee**. Automorphic forms for GL(n) and their L-functions (d'après Taylor), cont'd.

**Note**. Some of us will be in Lausanne on that day so perhaps this lecture will need to be postponed to a more propitious day in January. - December 21.
**Michele Fornea**. Special Christmas lecture. Brauer induction and potential automorphy (d'après Taylor and Carayol)