Odd-dimensional cohomology with finite coefficients and roots of unity

TL;DR

This paper establishes a link between the triviality of Galois action on twisted odd-dimensional étale cohomology groups of smooth projective varieties with finite coefficients and the existence of primitive roots of unity in their fields of definition.

Contribution

It proves that trivial Galois action on certain cohomology groups implies the presence of roots of unity, connecting algebraic geometry with number theory.

Findings

Trivial Galois action implies existence of roots of unity.

Provides a cohomological criterion for roots of unity.

Inspired by Serre's exercises on the Mordell--Weil theorem.

Abstract

We prove that the triviality of the Galois action on the suitably twisted odd-dimensional \'etale cohomogy group of a smooth projective varietiy with finite coefficients implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serre's Lectures on the Mordell--Weil theorem.