Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

TL;DR

This paper investigates the Galois cohomology related to Fermat curves over cyclotomic fields, connecting it with ray class fields and Heisenberg extensions, and provides explicit computations for the case p=3.

Contribution

It introduces a new analysis of a Galois cohomology group associated with Fermat curves using ray class fields and determines a significant subquotient explicitly for p=3.

Findings

Identifies a large subquotient of the Galois cohomology group from Heisenberg extensions.

Performs explicit Magma computations for p=3 to fully determine the cohomology group.

Establishes a connection between Galois cohomology and ray class fields of cyclotomic fields.

Abstract

The absolute Galois group of the cyclotomic field $K={\mathbb Q}(\zeta_p)$ acts on the \'etale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determines it completely.