The inverse Galois problem for PSL2(Fp)

Abstract

We show that the simple group ${\mathrm{PSL}}_2\left({\mathbb{F}}_p\right)$ occurs as the Galois group of an extension of the rationals for all primes $p\ge 5$. We obtain our Galois extensions by studying the Galois action on the second étale cohomology groups of a specific elliptic surface.