Mod p Monodromy of Cyclic Covers of the Projective Line

In this paper, we prove a big monodromy theorem for the monodromy of cyclic coverings of projective line for cohomology with Fp-coefficients. This is a direct generalization of the results of Achter and Pries, where such a theorem is proved for cyclic coverings of degree 2 and 3. Instead of generalizing their methods, we adapt the proof of the analogous theorem for integral cohomology. In our subsequent work, we will apply this theorem to construct in infinitely many cases Galois extensions of Q with Galois group PSL(n, q) and PSU(n, q), where q can be an arbitrarilty large prime power.