Contemporary Mathematics Volume 649, 2015 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 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 1
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