On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varieties

We study the action of $\operatorname{Gal}(\overline k/k)$ on the ring $H^*(A,\mathbf A^f)$, where $A$ is an Abelian variety defined over the field $k$ of characteristic zero and $\mathbf A^f$ is the ring of finite adèles of the field of rational numbers. We prove that there exists a subgroup of finite index in $\operatorname{Gal}(\overline k/k)$ which acts as scalars on $R^p(A)\otimes_{\mathbf Q}\mathbf A^f$, where $R^p(A)\subset H^{2p}(A,\mathbf Q)$ is the space of rational cohomology classes of type $(p,p)$.
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