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Functoriality and the inverse Galois problem. (English) Zbl 1194.11062
Let \(\ell \) be a prime number and \(m,t\) natural numbers. Based on ideas of G. Wiese [“On projective linear groups over finite fields as Galois groups over the rational numbers”, Modular forms on Schiermonnikoog. Based on the conference on modular forms, Schiermonnikoog, Netherlands, October 2006. Cambridge: Cambridge University Press, 343–350 (2008; Zbl 1217.12004)], the authors prove that the finite simple group \(S:=\text{PSp}_{2m}({\mathbb F}_{\ell^k})\) is the Galois group of some finite Galois extension over the field of rational numbers for some multiple \(k\) of \(t\). They show the existence of a surjective continuous homomorphism from the absolute Galois group of \(\mathbb Q\) to \(S\) (for some suitable \(k\)) by making use of a self-dual automorphic representation \(\Pi\) of \(\text{GL}_n({\mathbb A}_{\mathbb Q})\) which is regular algebraic at infinity, where \(L(s,\Lambda^2\Pi)\) has a pole at \(s=1\) and where some restrictions on the ramification have been imposed. One of the main ingredients in exhibiting such a \(\Pi\) are the results in [J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi, Publ. Math., Inst. Hautes Etud. Sci. 99, 163–233 (2004; Zbl 1090.22010)]. Results on Langlands functoriality are also used in several other papers which in particular have been made use of in getting the crucial link between certain automorphic representations and Galois representations.

11F80 Galois representations
12F12 Inverse Galois theory
11F12 Automorphic forms, one variable
11R32 Galois theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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