Bounds for moments of $\ell$-torsion in class groups

TL;DR

This paper establishes new strong unconditional upper bounds on the moments of $ ext{ell}$-torsion in class groups of various number field extensions, improving previous results and providing new bounds even for quadratic fields.

Contribution

It provides the first unconditional bounds on the moments of $ ext{ell}$-torsion in class groups for a broad class of number fields, surpassing prior conditional and unconditional results.

Findings

Unconditional upper bounds for moments of $ ext{ell}$-torsion in class groups.

Improved bounds for counting degree $p$ $D_p$-extensions over $Q$.

Results applicable to degree $p$ and $n$ $S_n$-extensions of number fields.

Abstract

Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of degree $n$ $S_n$-extensions of $k$, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large $r$, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of $\mathbb{Q}$. When $r=1$, our results are new even for the family of all quadratic extensions of $\mathbb{Q}$, leading to an improved upper bound for the count of degree $p$ $D_p$-extensions over $\mathbb{Q}$ (where $D_p$ is the dihedral group of order $2p$).