On modular forms and the inverse Galois problem

Dieulefait, Luis; Wiese, Gabor

doi:10.1090/S0002-9947-2011-05477-2

On modular forms and the inverse Galois problem

Authors: Luis Dieulefait and Gabor Wiese
Journal: Trans. Amer. Math. Soc. 363 (2011), 4569-4584
MSC (2010): Primary 11F80; Secondary 12F12, 11F11
DOI: https://doi.org/10.1090/S0002-9947-2011-05477-2
Published electronically: April 11, 2011
MathSciNet review: 2806684
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Abstract: In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer , there is a positive density set of primes such that $\mathrm{PSL}_2(\mathbb{F}_{p^n})$ occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.

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