Vol. 14, No. 4, 2020

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

Jack Klys

Vol. 14 (2020), No. 4, 815–854

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.

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Cohen–Lenstra heuristics, arithmetic statistics, class groups, cyclic fields
Mathematical Subject Classification 2010
Primary: 11R29
Secondary: 11R37, 11R45
Received: 30 June 2017
Revised: 30 October 2019
Accepted: 27 November 2019
Published: 21 June 2020
Jack Klys
Department of Mathematics and Statistics
University of Calgary