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

JOURNAL OF ALGEBRA

卷:554

Article

Davis, Rachel¹  Pries, Rachel² 

[1] Univ Wisconsin Madison, Madison, WI USA

[2] Colorado State Univ, Ft Collins, CO 80523 USA

关键词: Cyclotomic field;    Class field theory;    Ray class field;    Absolute Galois group;    Heisenberg group;    Fermat curve;    Homology;    Galois cohomology;    Obstruction;    Transgression;   

DOI  :  10.1016/j.jalgebra.2020.02.030

来源: Elsevier

【 摘 要 】

The absolute Galois group of the cyclotomic field K = 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free