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arXiv · 1210.3299

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

Philipp Habegger
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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.

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Philipp Habegger. 2012-10-11. The Tate-Voloch Conjecture in a Power of a Modular Curve. https://arxiv.org/abs/1210.3299

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