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 Andr e-Oort conjecture (or the GRH) implies the following statement: for any closed subvariety X $Ag;1 over Q a , there exists a Weyl special point [(B; )] 2 Ag;1(Q a ) such that B is not isogenous to the abelian variety A underlying any point [(A; )] 2 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 spaceAg;1 replaced by a general Shimura variety.