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

2012/10/11 by Philipp Habegger, Habegger, Philipp · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1210.3299

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.

Cited by

• A lower bound for the distance between CM points on Shimura curves
• The p-adic distance of special points to subvarieties
• Torsion subschemes and the Tate-Voloch conjecture over p-adic fields

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