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

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 informati...

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Bibliographic Details
Published inJournal of algebra Vol. 554; pp. 78 - 105
Main Authors Davis, Rachel, Pries, Rachel
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.07.2020
Subjects
Absolute Galois group
Class field theory
Cyclotomic field
Fermat curve
Galois cohomology
Heisenberg group
Homology
Obstruction
Ray class field
Transgression
secondary
Ray class field
Absolute Galois group
Heisenberg group
Fermat curve
Class field theory
Transgression
Cyclotomic field
Homology
Galois cohomology
Obstruction
primary
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2020.02.030