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

TL;DR

This work proves a composite instance of the Cohen–Lenstra–Gerth heuristics by establishing an explicit asymptotic for the average 6-torsion in the class groups of quadratic fields, with a main term for imaginary fields and analogous real-field results. It also proves Malle’s conjecture for regular Galois $D_6$-extensions, delivering a leading constant that matches the Loughran–Santens prediction. The approach blends genus theory, the Delone–Faddeev correspondence for cubic fields, and an improved level of distribution for cubic-field discriminants, together with refined local-condition analysis via Shintani zeta functions and Hooley’s Delta tools. The results illuminate the independence of 2- and 3-torsion components in this setting and provide new nonabelian field-counts with explicit constants and error terms, advancing our understanding of arithmetic statistics in a nontrivial composite case.

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.

Figures (2)

• Figure 1: Current state of the art
• Figure 2: Our argument

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Remark
• Lemma 2.2
• Lemma 2.3
• proof