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ISSN 1088-6850(online) ISSN 0002-9947(print)



Algebraic cycles and the Hodge structure of a Kuga fiber variety

Author: B. Brent Gordon
Journal: Trans. Amer. Math. Soc. 336 (1993), 933-947
MSC: Primary 14C30; Secondary 14C25, 14F20, 14K30
MathSciNet review: 1097167
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Abstract: Let $ \tilde A$ denote a smooth compactification of the $ k$-fold fiber product of the universal family $ {A^1} \to M$ of elliptic curves with level $ N$ structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology $ {H^{\ast} }(\tilde A,\mathbb{Q})$ of $ \tilde A$ , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for $ \tilde A$ ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate's conjecture concerning the algebraic cycles in the étale cohomology $ H_{{\text{et}}}^{\ast} (\tilde A \otimes \bar{\mathbb{Q}},{\mathbb{Q}_l})$. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology $ {H^{\ast} }({E^k},\mathbb{Q})$ of the $ k$-fold product of an elliptic curve $ E$ without complex multiplication, and a verification of the generalized Hodge conjecture for $ {E^k}$ .

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Keywords: Kuga variety, elliptic curve, Hodge conjecture, generalized Hodge conjecture, Abel-Jacobi map, intermediate Jacobian, Tate conjecture, cusp forms
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