We compute all the moments of the -torsion in the first step of a filtration of the class group defined by Gerth (1987) for cyclic fields of degree , unconditionally for 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 case this gives the distribution of the -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 -torsion in quadratic fields.
"The distribution of $p$-torsion in degree $p$ cyclic fields." Algebra Number Theory 14 (4) 815 - 854, 2020. https://doi.org/10.2140/ant.2020.14.815