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Hodge structures on abelian varieties of type IV. (English) Zbl 1086.14008

The author pursues his study of the inter-relations between the usual and Grothendieck’s general Hodge conjectures, in the case of an abelian variety \(A\) of PEL type [see, for instance, his earlier paper, Ann. Math. (2) 155, No. 3, 915–928 (2002; Zbl 1073.14014) for type III]. He here assumes that \(\text{End}_Q(A)\) is a quadratic imaginary field \(k\), fixes an elliptic curve \(E\) with CM by \(k\), and shows that the usual conjecture for products of \(A\) with a power of \(E\) implies the general one for all products \(A^n\times E^m\), and in particular, for all powers of \(A\). The discussion depends on the signature \((p,q)\) of the hermitian form induced by a polarization on the Betti homology. As an application, he deduces from the work of K. A. Ribet [Am. J. Math. 105, 523–538 (1983; Zbl 0586.14003)], C. Schoen [Compos. Math. 114, No. 3, 329–336 (1998; Zbl 0926.14002)], and K. Koike on the algebraicity of Weil cycles, several cases where \(A\) and its powers satisfy the general Hodge conjecture, e.g. if \((p,q)=(3,2)\) and \(k\) has non trivial units.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14K05 Algebraic theory of abelian varieties
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