On the galois action on rational cohomology classes of type (p, p) of abelian varieties

M. V. BorovoǏ

doi:10.1070/SM1974v023n04ABEH002185

On the galois action on rational cohomology classes of type (p, p) of abelian varieties

M. V. BorovoǏ

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

TWe study the action of Gal(k/k) on the ring H* (A, A^f), where A is an abelian variety defined over the field k of characteristic zero and 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 Gal(K/K) which acts as scalars on R^p(A)⊗_Q A^fwhere R_p(A)H^2P ⫅(A, Q) is the space of rational cohomology classes of type (P, P).

Original language	English
Pages (from-to)	613-616
Number of pages	4
Journal	Mathematics of the USSR - Sbornik
Volume	23
Issue number	4
DOIs	https://doi.org/10.1070/SM1974v023n04ABEH002185
State	Published - 30 Apr 1974

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abstract = "TWe study the action of Gal(k/k) on the ring H* (A, Af), where A is an abelian variety defined over the field k of characteristic zero and Af is the ring of finite ad{\`e}les of the field of rational numbers. We prove that there exists a subgroup of finite index in Gal(K/K) which acts as scalars on Rp(A)⊗Q Afwhere Rp(A)H2P ⫅(A, Q) is the space of rational cohomology classes of type (P, P).",

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N2 - TWe study the action of Gal(k/k) on the ring H* (A, Af), where A is an abelian variety defined over the field k of characteristic zero and Af is the ring of finite adèles of the field of rational numbers. We prove that there exists a subgroup of finite index in Gal(K/K) which acts as scalars on Rp(A)⊗Q Afwhere Rp(A)H2P ⫅(A, Q) is the space of rational cohomology classes of type (P, P).

AB - TWe study the action of Gal(k/k) on the ring H* (A, Af), where A is an abelian variety defined over the field k of characteristic zero and Af is the ring of finite adèles of the field of rational numbers. We prove that there exists a subgroup of finite index in Gal(K/K) which acts as scalars on Rp(A)⊗Q Afwhere Rp(A)H2P ⫅(A, Q) is the space of rational cohomology classes of type (P, P).

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On the galois action on rational cohomology classes of type (p, p) of abelian varieties

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