Abelian varieties isogenous to a Jacobian

Abstract

We define a notion of Weyl CM points in the moduli space A g,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⫋A g,1 over Q a , there exists a Weyl special point [(B,μ)]∈A g,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 A g,1 replaced by a general Shimura variety.