The Galois action and cohomology of a relative homology group of Fermat Curves

TL;DR

This paper provides explicit formulas for the Galois group action on the homology of Fermat curves and investigates related invariants and cohomology groups to understand obstructions to rational points.

Contribution

It offers new explicit descriptions of Galois actions and cohomological invariants for Fermat curves under Vandiver's conjecture, extending previous work by Anderson.

Findings

Explicit formulas for Galois action on homology

Analysis of invariants related to rational points

Cohomology groups indicating obstructions

Abstract

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(\zeta_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.