Non-Archimedean Approximations by Special Points

Tate and Voloch proved that a linear form in roots of unity is either zero or p-adically bounded from below by a positive constant that is independent of the roots of unity. They also conjectured that a subvariety of a semi-abelian variety does not admit arbitrarily good p-adic approximations by torsion points. Buium and Mattuck made progress and Scanlon gave a full proof using work of Chatzidakis and Hrushovski on the model theory of difference fields.

I will present some work in progress towards a modular version of the Tate-Voloch Conjecture. In a product of classical modular curves, a special point of ordinary reduction at p cannot p-adically approximate a fixed sub- variety to well. The proof relies on recent joint work with Pila on modular Mordell-Lang and a refinement of it by Pila.