Non-parametric sets of regular realizations over number fields

TL;DR

This paper demonstrates that for many finite groups, Galois extensions over a number field cannot all be derived from finitely many regular Galois extensions via specialization, highlighting limitations in realizing certain groups through specialization.

Contribution

It establishes non-parametricity results for various groups, showing that not all Galois extensions can be obtained from finitely many regular extensions, and extends this to infinitely many extensions under a conjectural assumption.

Findings

Certain groups' Galois extensions are not obtainable by specialization from finitely many regular extensions.

Examples include abelian, dihedral, symmetric, and linear groups over finite fields.

A similar non-parametricity result holds for infinitely many extensions under a conjectural theorem.

Abstract

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular of a certain type, under a conjectural "uniform Faltings" theorem".