Abelian varieties isogenous to a Jacobian

Abstract

We define a notion of Weyl CM points in the moduli space $\mathcal{A}_{g,1}$ 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 $X\subsetneqq \mathcal{A}_{g,1}$ over $\mathbb{Q}^{\rm a}$, there exists a Weyl special point $[(B,\mu)]\in \mathcal{A}_{g,1}(\mathbb{Q}^{\rm a})$ such that $B$ is not isogenous to the abelian variety $A$ underlying any point $[(A,\lambda)]\in X$. The title refers to the case when $g\geq 4$ 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 $\mathcal{A}_{g,1}$ replaced by a general Shimura variety.