Cyclic covers of the projective line and PSL(n,q) as Galois groups

Despite a lot of progress, the question of whether all the finite groups PSL(n,q) of all n by n matrices with coefficients in F_q and of determinant 1, modulo center, occur as Galois groups of a Galois extension of Q remains open. We study the cohomology of curves with an automorphism with F_p-coefficients, for which this automorphism naturally makes the cohomology group into an F_q-vector space. We give explicit congruence conditions under which the Galois action on the cohomology group has image exactly PSL(n,q). The results typically contain congruence conditions on both n and q. For example, we prove that if p is a prime congruent to -1 modulo 5, and n is an odd integer, not divisible by 5, then PSL(2n,p^2) is a Galois group over Q.