# Non-archimedean Approximations by Special Points

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Philipp Habegger , IAS
Fine Hall 214

Let x_1,x_2,... be a sequence of n-tuples of roots of unity and suppose X is a subvariety of the algebraic torus. For a prime number p, Tate and Voloch proved that if the p-adic distance between x_k and X tends to 0 then all but finitely many sequence members lie on X. Buium and Scanlon later generalized this result. The distribution of those x_k that lie on X is governed by the classical (and resolved) Manin-Mumford Conjecture. I will present a modular variant of Tate and Voloch's discreteness result. It was motivated by the analogy between the conjectures of Manin-Mumford and Andre-Oort.