---
abstract: 'We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.'
address:
- 'Department of Mathematics, Imperial College London'
- 'Department of Mathematics, Imperial College London'
author:
- Kevin Buzzard
- Toby Gee
bibliography:
- 'buzzgee.bib'
title: The conjectural connections between automorphic representations and Galois representations
---

Introduction.
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Given an algebraic Hecke character for a number field $F$, a classical construction of Weil produces a compatible system of 1-dimensional $\ell$-adic representations of ${\operatorname{Gal}}(\overline{F}/F)$. In the late 1950s, Taniyama’s work [@MR0095161] on $L$-functions of abelian varieties with complex multiplications led him to consider certain higher-dimensional compatible systems of Galois representations, and by the 1960s it was realised by Serre and others that Weil’s construction might well be the tip of a very large iceberg. Serre conjectured the existence of 2-dimensional $\ell$-adic representations of ${\operatorname{Gal}}({\overline{{\mathbb{Q}}}}/{\mathbb{Q}})$ attached to classical modular eigenforms for the group ${\operatorname{GL}}_2$ over ${\mathbb{Q}}$, and their existence was established by Deligne not long afterwards. Moreover, Langlands observed that one way to attack Artin’s conjecture on the analytic continuation of Artin $L$-functions might be via first proving that any non-trivial $n$-dimensional irreducible complex representation of the absolute Galois group of a number 