The distribution of p-torsion in degree p cyclic fields

Klys, Jack

doi:10.2140/ant.2020.14.815

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The distribution of $p$-torsion in degree $p$ cyclic fields

Jack Klys

Vol. 14 (2020), No. 4, 815–854

DOI: 10.2140/ant.2020.14.815

Abstract

We compute all the moments of the $p$ -torsion in the first step of a filtration of the class group defined by Gerth (1987) for cyclic fields of degree $p$ , unconditionally for $p = 3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen–Lenstra–Martinet conjectures. In the $p = 3$ case this gives the distribution of the $3$ -torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Klüners (2007) in their proof of the distribution of the $4$ -torsion in quadratic fields.

Keywords

Cohen–Lenstra heuristics, arithmetic statistics, class groups, cyclic fields

Mathematical Subject Classification 2010

Primary: 11R29

Secondary: 11R37, 11R45

Milestones

Received: 30 June 2017

Revised: 30 October 2019

Accepted: 27 November 2019

Published: 21 June 2020

Authors

Jack Klys
Department of Mathematics and Statistics University of Calgary Canada

Vol. 14, No. 4, 2020