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Serre’s conjecture and its consequences

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The seventh Takagi Lectures

November 22, 2009 (Sun) 10:00–11:00 November 23, 2009 (Mon) 11:00–12:00 Graduate School of Mathematical Sciences The University of Tokyo

Serre’s conjecture and its consequences

Chandrashekhar Khare

(University of California, Los Angeles)

Abstract

I will give a historically motivated account of Serre’s conjecture about mod p rep- resentations of the absolute Galois group of the rationals. This was proved by J.-P.

Wintenberger and myself, together with a certain input of Kisin.

The context in which Serre made his conjecture was the work of Serre and Swinnerton- Dyer which explained the congruences Ramanujan had found for the Ramanujan τ-function. The explanation was via the study of images of Galois representations Deligne attached to the Ramanujan Δ-function (again conjectured by Serre). Here Δ(z) = qΠ(1−qn)24 =

nτ(n)qn with q = e2πiz.

I will also explain some of the consequences of Serre’s conjecture. For instance it implies Artin’s conjecture for 2-dimensional, complex, odd representations of the absolute Galois group of the rationals.

I will also talk about the ideas of the proof and some questions they lead to.

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