Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number fields, conditional on standard $L$-function conjectures

TL;DR

This paper establishes new conditional bounds on the m-torsion parts of class groups of number fields for m=2, 3, 4, 5, by relating them to Selmer groups of elliptic curves and using the refined BSD conjecture.

Contribution

It introduces a novel approach linking class group torsion to elliptic curve Selmer groups under standard L-function conjectures, extending bounds to multiple torsion cases.

Findings

Conditional bounds on 2-5 torsion in class groups of number fields.

Relates class group torsion to Selmer groups of elliptic curves.

Uses refined BSD conjecture for global estimates.

Abstract

We prove new conditional bounds on the the $m$-torsion of class groups of number fields of any fixed degree, for $m=2$, $3$, $4$, and $5$. Our methods first recast the problem in the language of class groups of Galois modules, which allows us to relate these torsion subgroups to Selmer groups of elliptic curves. We then obtain a global estimate using the refined BSD conjecture, in a similar way to how one normally uses the Brauer-Siegel bound. Our methods are potentially very general, but rely on the existence of motives with very special $\mathbb{Z}/m\mathbb{Z}$-cohomology. In particular, the restriction to $m=2$, $3$, $4$, and $5$ stems from needing an elliptic curve over $\mathbb{Q}$ with $m$-torsion subgroup isomorphic to $\mathbb{Z}/m\mathbb{Z}\oplus\mu_m$.