Algebraic cycles on a simple \(4P\)-dimensional abelian variety over a number field
type: Item
instance of: scholarly article
title: Algebraic cycles on a simple \(4P\)-dimensional abelian variety over a number field
zbMATH Open document ID: 0934.14003
DOI: 10.1007/BF02433954
author: S. G. Tankeev
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-10-22T00:00:00Z
review text: Let \(J\) be an abelian variety over a number field \(k\subset\mathbb C\) and set \(J_{\mathbb C}=J\otimes_k\mathbb C\). There are a number of conjectures on algebraic cycles of \(J_{\mathbb C}\) (which can be stated even in a more general setting) such as: the Hodge conjecture, the general Hodge-Grothendieck conjecture, the Tate conjecture and the Mumford-Tate conjecture. For example, the Hodge conjecture asserts that the \(\mathbb Q\)-space \(H^{2r}(J_{\mathbb C},\mathbb Q)\cap H^{r,r}\) is generated by the cohomology classes of algebraic cycles of dimension \(r\) on \(J_{\mathbb C}\), and the Tate conjecture asserts that for every prime \(l\), the \(\mathbb Q_l\)-space \(H^{2r}_{et}(J\otimes_k\overline{k},\mathbb Q_l(r))^ {\text{Gal}(\overline{k}|k)}\) is generated by the cohomology classes of algebraic cycles of codimension \(r\) on \(J\). In the paper under review the author proves four interesting theorems which basically assert that (under certain additional hypotheses) these conjectures hold true for a simple \(4p\)-dimensional abelian variety \(J\) over \(k\) (where \(p\) is a prime natural number).
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 1213212
zbMATH Keywords: simple abelian varieties
zbMATH Keywords: algebraic cycles
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: general Hodge-Grothendieck conjecture
zbMATH Keywords: Tate conjecture
zbMATH Keywords: Mumford-Tate conjecture
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
cites work: Q3900179
cites work: Q3871018
cites work: Q5637018
cites work: Q3955498
cites work: Q3803260
cites work: ALGEBRAIC CYCLES ON AN ABELIAN VARIETY WITHOUT COMPLEX MULTIPLICATION
cites work: Cycles on Abelian varieties and exceptional numbers
cites work: Q5532660
cites work: Q3971179
cites work: On the l-adic representations attached to simple abelian varieties of type IV
cites work: ℓ-Adic and λ-Adic Representations Associated to Abelian Varieties Defined Over Number Fields
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: Variétés abeliennes ordinaires sur un corps fini
cites work: Hodge's general conjecture is false for trivial reasons
cites work: Q4307191
cites work: Some Recent Developments in the Theory of Algebraic Varieties
cites work: Q5628266
cites work: Q3890896
cites work: On analytic families of polarized abelian varieties and automorphic functions
cites work: Q5615218
cites work: Abelian varieties having a reduction of \(K\)3 type
full work available at URL: https://doi.org/10.1007/bf02433954
OpenAlex ID: W2002875646
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description: scientific article; zbMATH DE number 1213212