Abelian varieties isogenous to a Jacobian

Abstract

We dene a notion of Weyl CM points in the moduli space Ag,1 of g-dimensional principally polarized abelian varieties and show that the André-Oort conjecture (or the GRH) implies the following statement: for any closed subvariety X ⊂≠ Ag,1 over ℚa, there exists a Weyl special point [(B, μ)] ∈ Ag,1 (ℚa) such that B is not isogenous to the abelian variety A underlying any point [(A, λ)] ∈ X. The title refers to the case when g ≥ 4 and X is the Torelli locus; in this case Tsimerman has proved the statement unconditionally. The notion of Weyl special points is generalized to the context of Shimura varieties, and we prove a corresponding conditional statement with the ambient space Ag,1 replaced by a general Shimura variety.