#### 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
##### 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