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.

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      author={Tsimerman, J.},
      TITLE={The existence of an abelian variety over {$\overline{\mathbb{Q}}$} isogenous to no Jacobian},
      JOURNAL={Ann. of Math.},
      VOLUME={176},
      PAGES={637---650},
      DOI = {10.4007/annals.2012.176.1.12},
      YEAR={2012},
     }
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    @misc{Ullmo.Yaf,
      author={Ullmo, E. and Yafaev, A.},
      TITLE={Galois orbits of special {S}himura varieties},
      NOTE={[to appear]},
     }
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    @article {Weil-76, MRKEY = {0422164},
      AUTHOR = {Weil, Andr{é}},
      TITLE = {Sur les périodes des intégrales abéliennes},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {29},
      YEAR = {1976},
      NUMBER = {6},
      PAGES = {813--819},
      ISSN = {0010-3640},
      MRCLASS = {10D20 (14K25 12A70 10D25)},
      MRNUMBER = {0422164},
      MRREVIEWER = {Neal Koblitz},
      ZBLNUMBER = {0342.14020},
      DOI = {10.1002/cpa.3160290620},
     }

Authors

Ching-Li Chai

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

Frans Oort

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