Fourier transform for \'etale motivic cohomology

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.