Asymptotics for $6$-torsion and $D_6$-extensions

TL;DR

This paper proves asymptotic formulas for the average 6-torsion in class groups of quadratic fields and confirms Malle's conjecture for Galois D6-extensions, advancing understanding of number field distributions.

Contribution

It establishes the first asymptotic for 6-torsion in quadratic class groups and verifies Malle's conjecture for D6-extensions, filling gaps in number field heuristics.

Findings

Asymptotic formula for average 6-torsion in quadratic class groups

Proof of Malle's conjecture for Galois D6-extensions

Advancement in Cohen--Lenstra--Gerth heuristics

Abstract

We prove a composite case of the Cohen--Lenstra--Gerth heuristics. Specifically, we establish an asymptotic for the average $6$-torsion of the class group of quadratic number fields. We also prove Malle's conjecture for Galois $D_6$-extensions.