Abelian varieties isogenous to a Jacobian

We define 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 Q a , there exists a Weyl special point [(B, µ)] ∈ Ag,1(Q a ) 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.