Mod p Monodromy of Cyclic Covers of the Projective Line

TL;DR

This paper proves a big monodromy theorem for cyclic covers of the projective line with Fp-coefficients, extending previous results for degrees 2 and 3, and sets the stage for constructing Galois extensions with specific groups.

Contribution

It generalizes monodromy results to arbitrary degrees by adapting proofs from integral cohomology, enabling future applications in Galois theory.

Findings

Established big monodromy theorem for cyclic covers with Fp-coefficients.

Extended previous degree-specific results to arbitrary degrees.

Set groundwork for constructing Galois extensions with PSL(n, q) and PSU(n, q).

Abstract

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.