Fourier-stable subrings in the Chow rings of abelian varieties

We study subrings in the Chow ring ${\text{CH}}^{\ast }(A{)}_{\mathbb{Q}}$ of an abelian variety $A$, stable under the Fourier transform with respect to an arbitrary polarization. We prove that by taking Pontryagin products of classes of dimension $\le 1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in ${\text{CH}}^{\ast }(A{)}_{\mathbb{Q}}$. Another result concerns the relation between the Pontryagin product and the usual product on the ${\text{CH}}^{\ast }(A{)}_{\mathbb{Q}}$. We prove that the operator of the usual product with a cycle is a differential operator with respect to the Pontryagin product and compute its order in terms of the Beauville’s decomposition of ${\text{CH}}^{\ast }(A{)}_{\mathbb{Q}}$.