Arithmetic descent of specializations of Galois covers

TL;DR

This paper investigates when specializations of Galois covers over number fields can be descended to the rational numbers, showing positive results under certain group conditions and providing explicit counterexamples.

Contribution

It establishes conditions under which Galois specializations descend arithmetically to , including new results for cyclic groups and cases requiring base change.

Findings

Positive descent results for regularly realizable groups after base change

Descent for cyclic groups without base change

Explicit example of a cover with no rational descent points

Abstract

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the $G$-Galois field extension induced by specialization "arithmetically descends" to $\mathbb{Q}$ (i.e., there exists a $G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if one first allows a base change to a finite extension of $K$. If one does not allow base change, we prove that the answer is positive when $G$ is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of $\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.