On the $p$-rank of class groups of $p$-extensions

TL;DR

This paper develops a local-global principle for embedding problems in global fields, enabling the derivation of bounds on the $p$-rank of class groups of certain $p$-extensions using Galois group structures.

Contribution

It introduces a new local-global approach to describe maximal pro-$p$ Galois groups with restricted ramification and applies this to bound class group ranks in specific extensions.

Findings

Established bounds for the $p$-torsion part of class groups of $p$-extensions.

Provided explicit bounds depending on Galois and inertia subgroup structures.

Analyzed $p$-rank of class groups in cyclic $p$-extensions and multiquadratic extensions of $bQ$.

Abstract

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k=\mathbb{Q}$ and $k=\mathbb{F}_q(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-torsion group of the class group of $K$, and these bounds depend only on the structure of the Galois group and the inertia subgroups of $K/k$. Finally, we study the $p$-rank of class groups of cyclic $p$-extensions of $\mathbb{Q}$ and the $2$-rank of class groups of multiquadratic extensions of $\mathbb{Q}$, for a fixed ramification type.