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 Mat. Sb. (N.S.), 1974, Volume 94(136), Number 4(8), Pages 649–652 (Mi msb3737)

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

M. V. Borovoi

Abstract: 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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English version:
Mathematics of the USSR-Sbornik, 1974, 23:4, 613–616

Bibliographic databases:

UDC: 519.4+513.836
MSC: Primary 14K15, 14G13; Secondary 14G25, 14C30

Citation: M. V. Borovoi, “On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varieties”, Mat. Sb. (N.S.), 94(136):4(8) (1974), 649–652; Math. USSR-Sb., 23:4 (1974), 613–616

Citation in format AMSBIB
\Bibitem{Bor74} \by M.~V.~Borovoi \paper On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varieties \jour Mat. Sb. (N.S.) \yr 1974 \vol 94(136) \issue 4(8) \pages 649--652 \mathnet{http://mi.mathnet.ru/msb3737} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=352101} \zmath{https://zbmath.org/?q=an:0326.14013} \transl \jour Math. USSR-Sb. \yr 1974 \vol 23 \issue 4 \pages 613--616 \crossref{https://doi.org/10.1070/SM1974v023n04ABEH002185} 

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This publication is cited in the following articles:
1. M. V. Borovoi, “Skhemy $M_C(G,h)$ i gruppa Mamforda–Teita”, UMN, 32:6(198) (1977), 245–246
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