The Galois Action on Consistent Maps

TL;DR

This paper investigates the Galois action on consistent maps related to a specific vector space over algebraic numbers, establishing conditions for invariance and generalizing previous results on rational-valued maps.

Contribution

It introduces a Galois action on consistent maps and determines when these maps are invariant, extending earlier work on rational-valued maps over non-Archimedean places.

Findings

Galois action on consistent maps is characterized.

Conditions for invariance under Galois action are established.

Generalization of previous rational-valued map results.

Abstract

A 2009 article of Allcock and Vaaler explored of the $\mathbb Q$-vector space $\mathcal G := \overline{\mathbb Q}^\times/{\overline{\mathbb Q}^\times_{\mathrm{tors}}}$, showing how to represent it as part of a function space on the places of $\overline{\mathbb Q}$. Several years later, the author began attempts to examine dual spaces related to $\mathcal G$ in an effort to obtain Riesz-type representation theorems. Those results required the construction of an object called a {\it consistent map}. We study a natural Galois action on consistent maps and establish when consistent maps are invariant under this action. Our results generalize earlier work of the author regarding rational valued consistent maps over non-Archimedean places of $\mathbb Q$.