Pointwise Bound for $\ell$-torsion in Class Groups: Elementary Abelian Extensions

TL;DR

This paper establishes a new, unconditional pointwise upper bound on the -torsion in class groups of elementary abelian extensions, surpassing previous bounds under certain conditions, and does so without assuming GRH.

Contribution

It provides the first unconditional, pointwise bounds on -torsion in class groups for elementary abelian extensions, improving upon prior results under specific parameters.

Findings

Unconditional bounds on -torsion in class groups for elementary abelian extensions.

Bounds surpass previous results when the rank is sufficiently large.

Results are pointwise, not average, and do not rely on GRH.

Abstract

Elementary abelian groups are finite groups in the form of $A=(\mathbb{Z}/p\mathbb{Z})^r$ for a prime number $p$. For every integer $\ell>1$ and $r>1$, we prove a non-trivial upper bound on the $\ell$-torsion in class groups of every $A$-extension. Our results are pointwise and unconditional. When $r$ is large enough, the pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg-Venkatesh under GRH.