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

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 $\mathbb{Z}/m\mathbb{Z}\oplus\mu_m$.