The Tate-Voloch conjecture in a power of a modular curve

Let p be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be p-adically too close to a fixed subvariety without lying on it. The current work is motivated by the analogy between torsion points on semi-abelian varieties and special or CM points on Shimura varieties. We prove the analog of Tate and Voloch’s result in a power of the modular curve Y (1) on replacing torsion points by points corresponding to a product of elliptic curves with complex multiplication and ordinary reduction. Moreover, we show that the assumption on ordinary reduction is necessary.