Submission date: 2 May 2019

Abstract:

We prove weak versions of the Zilber-Pink conjecture in the semi-abelian and modular settings. Given a “small” set Γ, which is a subgroup of finite rank in the semi-abelian case and a subset of ℚ^{alg} consisting of special points and Hecke orbits of finitely many non-special points in the modular case, we consider Γ-special subvarieties---weakly special subvarities containing a point of Γ (or a tuple from Γ in the modular case)---and show that every variety V contains only finitely many maximal Γ-atypical subvarieties, i.e. atypical intersections of V with Γ-special varieties the weakly special closures of which are Γ-special. The Mordell-Lang conjecture and its modular analogue (established by Habegger and Pila), as well as the Ax-Schanuel theorem in each setting, play a key role in our proofs.

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Full text arXiv 1905.00827: pdf, ps.