On Modular Forms and the Inverse Galois Problem

TL;DR

This paper advances the inverse Galois problem by demonstrating the existence of many primes p for which PSL_2(F_{p^n}) appears as a Galois group, using modular forms and residual Galois representations.

Contribution

It establishes new cases of the inverse Galois problem by linking modular forms to Galois groups and constructing families with maximal residual Galois image.

Findings

Positive density of primes p with PSL_2(F_{p^n}) as Galois group

Construction of modular form families with large residual Galois images

Use of good-dihedral primes to control Galois representations

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.