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On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varieties
Sbornik. Mathematics, Tome 23 (1974) no. 4, pp. 613-616 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $(p,p)$. Bibliography: 6 titles. 
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     author = {M. V. Borovoi},
     title = {On the {Galois} action on rational cohomology classes of type $(p,p)$ of {Abelian} varieties},
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M. V. Borovoi. On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varieties. Sbornik. Mathematics, Tome 23 (1974) no. 4, pp. 613-616. http://geodesic.mathdoc.fr/item/SM_1974_23_4_a7/

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