Twisted covers and specializations

TL;DR

This paper investigates when a $k$-étale algebra can be realized as a specialization of a $k$-cover, introducing a generalized twisting lemma that simplifies the problem to finding rational points on associated varieties, with various applications.

Contribution

It generalizes and unifies existing twisting lemmas and applies them to new contexts like Hilbert's irreducibility, Tchebotarev theorems, and specialization properties over PAC or ample fields.

Findings

Unified a class of twisting lemmas for specialization problems.

Extended applications to number fields and finite fields.

Provided new criteria for specializations over PAC or ample fields.

Abstract

The central topic is this question: is a given $k$-\'etale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields.