Average bounds for the $\ell$-torsion in class groups of cyclic extensions

TL;DR

This paper establishes new bounds on the $ ell$-torsion in class groups of cyclic extensions, valid for almost all such fields, and provides asymptotic counts for these fields with specific properties.

Contribution

It introduces non-trivial bounds for $ ell$-torsion in class groups of cyclic extensions and counts fields with prescribed splitting behavior.

Findings

Bounds hold for almost all cyclic degree-$p$-extensions of any number field.

Asymptotic formulas for counting fields with given splitting conditions.

Results apply to arbitrarily large degree cyclic extensions.

Abstract

For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.