Cohomology and torsion cycles over the maximal cyclotomic extension

Rössler, D
Szamuely, T

1 July 2019

Journal:

Journal für die Reine und Angewandte Mathematik (Crelle's Journal)

Last Updated:

2020-10-05T02:29:08.133+01:00

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.

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