The local dimension of a finite group over a number field

TL;DR

This paper constructs a special G-extension over a number field that can specialize to all local G-extensions at almost all primes, linking to the concept of essential dimension and its arithmetic analogue.

Contribution

It introduces a method to construct G-extensions over a transcendence degree 2 field that specialize to all local extensions, and explores the Hilbert-Grunwald property in this context.

Findings

Constructed G-extensions that specialize to all local G-extensions at almost all primes.

Established the connection between these extensions and the essential dimension of G.

Demonstrated the extension has the Hilbert-Grunwald property when G has a generic extension.

Abstract

Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely many primes of $K$. If furthermore $G$ has a generic extension over $K$, we show that the extension $E/F$ has the so-called Hilbert-Grunwald property. These results are compared to the notion of essential dimension of $G$ over $K$, and its arithmetic analogue.