Fourier transform for \'etale motivic cohomology

TL;DR

This paper explores the Fourier transform in the context of étale motivic cohomology for abelian varieties, establishing a PD-structure over the positive degree étale Chow ring, advancing the theoretical framework.

Contribution

It introduces a PD-structure over the positive degree part of the étale Chow ring with respect to the Pontryagin product for abelian varieties.

Findings

Existence of a PD-structure over the positive degree étale Chow ring.

Extension of Fourier transform concepts to étale motivic cohomology.

Integration of ideas from Moonen, Polishchuk, Beckman, and de Gaay Fortman.

Abstract

In the present article, we study the integral aspects of the Fourier transform of an abelian variety $A$ over a field $k$, using \'etale motivic cohomology, following the ideas and theory given by Moonen, Polishchuk and later by Beckman and de Gaay Fortman. We prove that there exists a PD-structure over the positive degree part of the \'etale Chow ring $\text{CH}^{\text{\'et}}_{>0}(A)$ with respect to the Pontryagin product.