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arXiv · 1806.08352

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

Rachel Davis·Rachel Pries
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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.

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Rachel Davis, Rachel Pries. 2018-06-21. Cohomology groups of Fermat curves via ray class fields of cyclotomic fields. https://arxiv.org/abs/1806.08352

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