On modular forms and the inverse Galois problem

Abstract

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational nu mbers. 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 resid ual Galois representations are as large as a priori possible. Both results essentially use Khare’s a nd Wintenberger’s notion of gooddihedral primes. Particular care is taken in order to exclud e nontrivial inner twists. 2000 Mathematics Subject Classification: 11F80 (primary); 12F12, 11F11.