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The Tate conjecture for generic abelian varieties. (English) Zbl 0821.14006

Let \(A \to V\) be a Kuga fiber variety of Mumford’s Hodge type, defined over a finitely generated subfield of \(\mathbb{C}\), and let \(\eta\) be the generic point of \(V\). We show that any element of \(H^{2r}_{\acute et} (A_{\overline \eta}, \mathbb{Q}_ l) (r)\) which is invariant under \(\text{Gal} (\overline {k(\eta)}/E)\) for some finite extension \(E\) of \(k(\eta)\), is fixed by the semisimple part of the Hodge group of \(A_ \eta\). If \(A \to V\) satisfies the \(H_ 2\)-condition, then the Hodge and Tate conjectures are equivalent for \(A_ \eta\), and the Mumford-Tate conjecture is true.

MSC:

14C25 Algebraic cycles
14K10 Algebraic moduli of abelian varieties, classification
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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