The Galois Action on Consistent Maps

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 $\mathbb Q$.