Algebra & Number Theory

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

Jack Klys

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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.

Article information

Source
Algebra Number Theory, Volume 14, Number 4 (2020), 815-854.

Dates
Received: 30 June 2017
Revised: 30 October 2019
Accepted: 27 November 2019
First available in Project Euclid: 30 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1593482419

Digital Object Identifier
doi:10.2140/ant.2020.14.815

Mathematical Reviews number (MathSciNet)
MR4114057

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R37: Class field theory 11R45: Density theorems

Keywords
Cohen–Lenstra heuristics arithmetic statistics class groups cyclic fields

Citation

Klys, Jack. The distribution of $p$-torsion in degree $p$ cyclic fields. Algebra Number Theory 14 (2020), no. 4, 815--854. doi:10.2140/ant.2020.14.815. https://projecteuclid.org/euclid.ant/1593482419


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