Cohomology and torsion cycles over the maximal cyclotomic extension

TL;DR

This paper generalizes Ribet's theorem on torsion points of abelian varieties over maximal cyclotomic extensions, connecting it to Galois actions on étale cohomology and proposing conjectures for higher codimension torsion cycles.

Contribution

It extends Ribet's finiteness result to a broader cohomological context and introduces conjectures for higher codimension torsion cycles with supporting evidence.

Findings

Finiteness of Galois fixed points on étale cohomology with Q/Z coefficients.

Support for conjectural generalization in codimension 2.

An analogue of the theorem in positive characteristic.

Abstract

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension $K$ of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of $K$ acts with finitely many fixed points on the \'etale cohomology with $\bf Q/\bf Z$-coefficients of a smooth proper $\overline K$-variety defined over $K$. We also present a conjectural generalization of Ribet's theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.