Minimal fields of definition for Galois action

Abstract

Let $K$ be a field, let $G$ be a finite group, and let $\bar X\rightarrow \bar Y$ be a $G$-Galois branched cover of varieties over $K^{sep}$. Given a mere cover model $X\rightarrow Y$ of this cover over $K$, in Part I of this paper I observe that there is a unique minimal field $E$ over which $X\rightarrow Y$ becomes Galois, and I prove that $E/K$ is Galois with group a subgroup of $Aut(G)$. In Part II of this paper, by making the additional assumption that $K$ is a field of definition (i.e., that there exists \it some \rm Galois model over $K$), I am able to give an explicit description of the unique minimal field of Galois action for $X\rightarrow Y$. Namely, if there exists a $K$-rational point of $X$ above an unramified point $P\in Y(K)$ then $E$ is contained in the intersection of the specializations at $P$ in the various different $G$-Galois models of $\bar X\rightarrow \bar Y$ over $K$. Using the same proof mechanism, I observe a reverse version of "The Twisting Lemma", which asserts that the behavior of the $K$-rational points on the various mere cover models over $K$, and the behavior of the specializations on the various $G$-Galois models over $K$, are all governed by a single equivalence relation (independent of the model) on the $K$-rational points of the base variety.