Abelian varieties isogenous to a Jacobian


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.


Ching-Li Chai

Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, PA 19104-6395

Frans Oort

Mathematisch Institüt, Universiteit Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands