Algebraic Cycles and the Hodge Structure of a Kuga Fiber Variety

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 ${E^k}$ .