On modular forms and the inverse Galois problem

Abstract

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 PSL 2 (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.

Constructing Hilbert modular forms without exceptional primes

“…The result now follows exactly as in Theorem 5.4.ii of [Di11] by using the isomorphism mentioned above.…”

“…In this section we extend the definition of tamely dihedral representation of [Di11] to totally real fields in order to exclude complex multiplication and inner twists.…”

“…By using a Lemma of Dimitrov we are going to be able to construct Hilbert modular newforms of arbitrary weight, not only of weight 2 as in [Di11]. On the other hand, if F is a Galois field of odd degree, we will add an extra ingredient to our construction in order to avoid the possibility that the Hilbert modular newforms considered come from a base change.…”

Women in Numbers Europe

“…In fact, by considering compatible systems of Galois representations attached to certain automorphic forms, we know (cf. Wiese 2008; Dieulefait and Wiese 2011;Khare et al 2008;Arias-de-Reyna et al 2013) that these groups are Galois groups over Q, for infinitely many integers r and infinitely many primes`. More precisely, we have:…”

Galois representations and the tame inverse Galois problem

“…This strategy has already been used to address the Inverse Galois Problem for families of linear groups (cf. [35], [10], [11], [36], [12], [17]). In order to make use of techniques of this kind, it is essential to have explicit criteria to determine precisely the image of such a representation.…”