The inverse Galois problem for PSL_2(F_p)

TL;DR

This paper proves that for all primes p>3, the simple group PSL_2(F_p) can be realized as a Galois group over the rationals, using elliptic surface cohomology techniques.

Contribution

It establishes the inverse Galois problem for PSL_2(F_p) over the rationals for all primes p>3, employing novel cohomological methods.

Findings

PSL_2(F_p) occurs as Galois group over Q for all p>3

Galois action on elliptic surface cohomology is key

New techniques connect elliptic surfaces to inverse Galois problem

Abstract

We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific elliptic surface.