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

DOI10.1093/IMRN/RNT025zbMATH Open1296.11071arXiv1210.3299OpenAlexW1988062172WikidataQ123019386 ScholiaQ123019386MaRDI QIDQ2878730

P. Habegger

Publication date: 5 September 2014

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1210.3299

zbMATH Keywords

elliptic curveTate-Voloch conjecturemodular curveordinary reductionCM point

Mathematics Subject Classification ID

Local ground fields in algebraic geometry (14G20) Arithmetic aspects of modular and Shimura varieties (11G18) Arithmetic ground fields for abelian varieties (14K15)

Related Items (3)

The Manin-Mumford conjecture and the Tate-Voloch conjecture for a product of Siegel moduli spaces ⋮ Tate conjecture for a product of a Shimura curve and a Picard modular surface ⋮ Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

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