A note on asymptotic class number upper bounds in $p$-adic Lie extensions

TL;DR

This paper investigates upper bounds on the growth of class numbers in certain $p$-adic Lie extensions of number fields, generalizing previous results by relaxing the structure of the Galois group.

Contribution

It extends Lei's work by providing asymptotic upper bounds for class number growth in more general $p$-adic Lie extensions with specific subgroup structures.

Findings

Established asymptotic upper bounds for $p$-exponents of class groups

Generalized Lei's results to broader $p$-adic Lie groups

Provided conditions under which the bounds hold

Abstract

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish asymptotic upper bounds for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that $H\cong \mathbb{Z}_p$.