Recovering function fields from their integral $\ell$-adic cohomology with the Galois action

TL;DR

This paper demonstrates how the Galois action on integral ll-adic cohomology of higher-dimensional algebraic varieties over non-local fields can uniquely determine the function field and base field in dimensions three and higher.

Contribution

It shows that, in dimension 3, the Galois action on cohomology fully recovers the function field and base field, extending previous understanding of field reconstruction.

Findings

Galois action on cohomology parameterizes divisorial valuations.

In dimension 3, the function field and base field are completely determined.

The approach applies recent theorems of Pop to higher-dimensional cases.

Abstract

In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.