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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 studya Galois cohomology group which is valuable for measuring an obstruction for$K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ whichcontains the information needed for measuring this obstruction. We determine alarge subquotient of this Galois cohomology group which arises from Heisenbergextensions of $K$. For $p=3$, we perform a Magma computation with ray classfields, group cohomology, and Galois cohomology, which determines itcompletely.

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