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

Rachel Davis, Rachel Pries

Published 2018-06-21Version 1

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 $X$. The information needed for measuring this obstruction factors through a $2$-nilpotent extension of $K$. By studying Heisenberg extensions of $K$, we determine a large subquotient of this Galois cohomology group. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely.