  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.
