Galois action on the homology of Fermat curves

TL;DR

This paper investigates the Galois module structure of the homology of Fermat curves, providing explicit computations for prime degrees and exploring implications for rational points and Galois cohomology.

Contribution

It proves that the Galois group of the splitting field over _p is elementary abelian of rank (p+1)/2 under Vandiver's conjecture and explicitly computes homology groups as Galois modules.

Findings

Galois group of the splitting field is elementary abelian of rank (p+1)/2 for prime p under Vandiver's conjecture

Explicit basis for the Galois group allows complete computation of homology groups as Galois modules

Computed Galois cohomology groups related to obstructions to rational points

Abstract

In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as a Galois module for the action of the absolute Galois group $G_{\mathbb{Q}(\zeta_n)}$. In particular, when $n$ is an odd prime $p$, he shows that the action of $G_{\mathbb{Q}(\zeta_p)}$ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial $1-(1-x^p)^p$. If $p$ satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over $\mathbb{Q}(\zeta_p)$ is an elementary abelian $p$-group of rank $(p+1)/2$. Using an explicit basis for this Galois group, we completely compute the relative homology, the homology, and the homology of an open subset of the degree $3$ Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.