Duke Math. J. 57 (1988), 333-345
Math. Annalen 285 (1989), 289-295
A Kuga fiber variety f : A → V is an abelian scheme parametrized by an arithmetic variety and constructed from a symplectic representation of an algebraic group. The zeta function of A is then the product of L-functions associated to the sheaves Rbf∗Ql on V. In the first paper we show how absolute Hodge cycles induce relations between the zeta functions of two Kuga fiber varieties over the same base; however these relations are valid only over some undetermined extension of the field of definition. In the second paper, the fields of definition of the Hodge cycles are investigated, and it is shown that the relations between the zeta functions hold over the fields of definition of the canonical models.
Math. Annalen 294 (1992), 225-234
A lower bound for the field of definition of a complex algebraic variety X is given by Bot X, the strong bottom field; this is a field k such that for any automorphism σ of C, Xσ ≅ X if and only if σ is the identity on k. The principal result of this paper is that if A → V is a Kuga fiber variety defined by a Q-irreducible representation satisfying a certain rigidity condition, and if the generic fibers are principal abelian varieties, then Bot A is an abelian extension of Bot V. In fact, the Galois group of Bot A over Bot V is embedded into the class group of a maximal order in a simple algebra. We give an example of a family of 4-dimensional abelian varieties for which this Galois group is isomorphic to the Hilbert class group of Q(√(-23)).
Pacific J. Math. 165 (1994), 207-216
Let τ : X1 → X2 be a strongly equivariant holomorphic embedding of one bounded symmetric domain into another. We show that if σ is an automorphism of C, then τσ: X1σ → X2σ is also strongly equivariant.
Bull. London Math. Soc. 26 (1994), 417-421
Let A → V be a Kuga fiber variety of Mumford's Hodge type, defined over a finitely generated subfield of C, and let η be the generic point of
V. We show that any element of
which is invariant under
for some finite extension E of k(η), is fixed by
the semisimple part of the Hodge group of Aη. If A → V
satisfies the H2-condition, then the Hodge and Tate conjectures are
equivalent for Aη, and the Mumford-Tate conjecture is true.
Canadian J. Math. 46 (1994), 1121-1134
If the Hodge ∗-operator on the L2-cohomology of Kuga fiber varieties is algebraic, then the Hodge conjecture is true for all abelian varieties.
Compositio Math. 109 (1997), 341-355
We investigate the relation between the usual and general Hodge conjectures, and prove that for a large class of abelian varieties A, the usual Hodge conjecture for all powers of A implies the general Hodge conjecture for A.
Internat. J. Math. 10 (1999), 667-675
We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.
We prove the general Hodge conjecture for any complex abelian variety of CM-type such that the Hodge ring of each power of the abelian variety is generated by divisors.
Last revision: July 25, 2005.