Divided powers on abelian varieties

TL;DR

This paper establishes the existence of divided powers in étale Chow groups of abelian varieties over separably closed fields, providing an integral lift of the Fourier transform away from characteristic and 2-torsion.

Contribution

It introduces new techniques to lift Deninger-Murre Chow-Künneth projectors integrally, enabling the construction of divided powers and Fourier transform lifts.

Findings

Existence of divided powers in étale Chow groups.

Integral lift of Fourier transform away from characteristic and 2-torsion.

New methods for lifting Chow-Künneth projectors.

Abstract

We prove the existence of divided powers in \'etale Chow groups of abelian varieties over a separably closed field, and hence of an integral lift of the Fourier transform, away from the characteristic and up to $2$-torsion. The method is to lift the Deninger-Murre Chow-K\"unneth projectors to integral ones, and draw consequences. Several techniques used here are new.