The Tate-Voloch Conjecture in a Power of a Modular Curve

TL;DR

This paper extends the Tate-Voloch conjecture to powers of the modular curve, demonstrating that CM points with ordinary reduction cannot be too close to a fixed subvariety unless they lie on it, highlighting the analogy with torsion points.

Contribution

It proves an analog of the Tate-Voloch result for CM points on powers of the modular curve, emphasizing the necessity of the ordinary reduction condition.

Findings

CM points with ordinary reduction cannot be p-adically too close to a fixed subvariety unless on it

The analogy between torsion points and CM points is reinforced in this setting

The ordinary reduction assumption is shown to be essential

Abstract

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.