Hodge structures on abelian varieties of type IV

Abstract

Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties. (2000): Primary 14C30, 14K20. Mathematics Subject Classification

Hodge Structures Associated toSU(p, 1)

“…In a series of articles [2][3][4][5][6][7][8] we have shown for a large class of abelian varieties that every effective Tate twist of a Hodge structure contained in the cohomology of one of these abelian varieties is isomorphic to a Hodge structure occurring in the cohomology of some abelian variety. Our earlier results apply to abelian varieties 4210 ABDULALI of type IV in only a few cases-namely, when the Hodge group is semisimple [2], or when the abelian variety is of CM-type [7], or, when the semisimple part of the Hodge group is a product of groups of the form SU p + 1 p [8].…”