Contemporary Mathematics

Volume 649, 2015

http://dx.doi.org/10.1090/conm/649/13016

Automorphic Galois Representations

and the Inverse Galois Problem

Sara Arias-de-Reyna

Abstract. A strategy to address the inverse Galois problem over Q consists

of exploiting the knowledge of Galois representations attached to certain auto-

morphic forms. More precisely, if such forms are carefully chosen, they provide

compatible systems of Galois representations satisfying some desired proper-

ties, e.g. properties that reflect on the image of the members of the system.

In this article we survey some results obtained using this strategy.

1. Introduction

The motivation for the subject of this survey comes from Galois theory. Let

L/K be a field extension which is normal and separable. To this extension one can

attach a group, namely the group of field automorphisms of L fixing K, which is

denoted as Gal(L/K). The main result of Galois theory, which is usually covered

in the program of any Bachelor’s degree in Mathematics, can be stated as follows:

Theorem 1.1 (Galois). Let L/K be a finite, normal, separable field extension.

Then there is the following bijective correspondence between the sets:

E field

K ⊆ E ⊆ L

←→

H ⊆ Gal(L/K)

subgroup

,

E −→ Gal(L/E)

LH

←− L

Usually the students are asked exercises of the following type: Given some

finite field extension L/Q which is normal, compute the Galois group Gal(L/Q)

attached to it. But, one may also ask the inverse question (hence the name inverse

Galois problem): Given a finite group G, find a finite, normal extension L/Q with

Gal(L/Q) G. This is not a question one usually expects a student to solve!

In fact, there are (many) groups G for which it is not even known if such a field

extension exists.

2010 Mathematics Subject Classification. Primary 11F80, 12F12.

This article is an expanded version of the plenary lecture I delivered at the conference Quintas

Jornadas de Teor´ ıa de N´ umeros (July 2013). I would like to thank the scientific committee for

giving me the oportunity to participate in this conference, and the organising committee for their

excellent work. I also want to thank Gabor Wiese and the anonymous referee for their remarks

and suggestions on a previous version of this article.

c 2015 American Mathematical Society

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