Cohomology and torsion cycles over the maximal cyclotomic extension

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 étale cohomology with $Q/\mathbf{Z}$-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.

Final version. Theorem 1.9 improved thanks to referee