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On projective linear groups over finite fields as Galois groups over the rational numbers. (English) Zbl 1217.12004
Edixhoven, Bas et al., Modular forms on Schiermonnikoog. Based on the conference on modular forms, Schiermonnikoog, Netherlands, October 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-49354-3/hbk). 343-350 (2008).
Let $$\ell$$ be a prime number. The author shows that infinitely many of the groups $$\text{PSL}_2(\mathbb F_{\ell^r})$$ occur as Galois groups of extensions of the rationals which are unramified outside $$\infty, \ell$$ and one further finite place.
He does so by using ideas from the papers of C. Khare and J.-P. Wintenberger on Serre’s modularity conjecture [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)] and even gets the information that the mentioned third place can be chosen from a certain set of primes with positive density.
For the entire collection see [Zbl 1149.11002].

##### MSC:
 12F12 Inverse Galois theory 11F80 Galois representations
##### Keywords:
Galois representation; level raising; Chebotarev density
Magma
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