Class groups of kummer extensions via cup products in Galois cohomology

Karl Schaefer; Eric Stubley

doi:10.1090/tran/7746

Class groups of kummer extensions via cup products in Galois cohomology

Karl Schaefer
, Eric Stubley

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We use Galois cohomology to study the p-rank of the class group of Q(N^1/p), where N ≡ 1 mod p is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5, we prove a complete characterization of the 5-rank of the class group of Q(N¹/⁵) in terms of whether or not Π _k ^(N ₌₁ ⁻1)/² k^k and √ ⁵ ₂ ⁻ ¹ are 5th powers mod N.

Original language	English
Pages (from-to)	6927-6980
Number of pages	54
Journal	Transactions of the American Mathematical Society
Volume	372
Issue number	10
DOIs	https://doi.org/10.1090/tran/7746
State	Published - Nov 15 2019

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title = "Class groups of kummer extensions via cup products in Galois cohomology",

abstract = "We use Galois cohomology to study the p-rank of the class group of Q(N1/p), where N ≡ 1 mod p is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5, we prove a complete characterization of the 5-rank of the class group of Q(N1/5) in terms of whether or not Π k (N =1 −1)/2 kk and √ 5 2 − 1 are 5th powers mod N.",

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AB - We use Galois cohomology to study the p-rank of the class group of Q(N1/p), where N ≡ 1 mod p is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5, we prove a complete characterization of the 5-rank of the class group of Q(N1/5) in terms of whether or not Π k (N =1 −1)/2 kk and √ 5 2 − 1 are 5th powers mod N.

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DO - 10.1090/tran/7746

M3 - Article

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 10

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Class groups of kummer extensions via cup products in Galois cohomology

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