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

Abstract

The absolute Galois group of the cyclotomic field K=Q(ζp) acts on the étale 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.