André–Pink–Mordell–Lang conjecture

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Let T=S×AT=S\times A be the product of a Shimura variety SS and an abelian variety AA, and let X⊊TX\varsubsetneq T be a weakly special generic closed irreducible subvariety. Fix a sub-Shimura variety S⊂Ag′S\subset\mathcal{A}_{g'} for some g′g', let s∈S(C)s\in S(\mathbb{C}), and let Σs\Sigma_s be the isogeny class of ss. Fix a subgroup of finite rank Γ≤A(C)\Gamma\leq A(\mathbb{C}). André–Pink–Mordell–Lang conjecture. The set

(Σs×Γ)∩X(\Sigma_s\times\Gamma)\cap X

is not Zariski dense in XX. This is an isogeny-class analogue of the preceding unlikely-intersection conjectures; the supplied text does not state a resolution.

References

Primary source

Gregorio Baldi, “On a conjecture of Buium and Poonen”, arXiv:1803.04946 (2019).

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