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

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.