Modular forms and some cases of the inverse Galois problem

TL;DR

This paper advances the inverse Galois problem by constructing new Galois extensions of the rationals with specific projective special linear groups as Galois groups using modular forms and residual Galois representations.

Contribution

It provides new cases of the inverse Galois problem by linking modular forms to Galois groups PSL_2 over finite fields for various primes.

Findings

Existence of Galois extensions with Galois group PSL_2(F_p) for all primes p.

Existence of Galois extensions with Galois group PSL_2(F_{p^3}) for certain primes p.

Specific construction methods using weight 3 newforms.

Abstract

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois group $PSL_2(\mathbb{F}_p)$ for all primes $p$ and $PSL_2(\mathbb{F}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}$.