André-Pink-Zannier conjecture for the universal family of abelian varieties

Let Ag\mathfrak{A}_g be the universal family of principally polarized abelian varieties, and let YY be a subvariety of Ag\mathfrak{A}_g. For sAgs\in\mathfrak{A}_g, let Σ\Sigma be the generalized Hecke orbit of ss. André-Pink-Zannier conjecture. If

YΣZar=Y,\overline{Y\cap\Sigma}^{\mathrm{Zar}}=Y,

then YY is weakly special. This is an important case of the Zilber-Pink conjecture for a mixed Shimura variety; its status is not resolved in the supplied source.

Sources & referencesView supporting material

Primary source

Ziyang Gao, “A special point problem of André-Pink-Zannier in the universal family of abelian varieties”, arXiv:1407.5578 (2015).

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