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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the explicit Torsion Anomalous Conjecture

Authors: S. Checcoli, F. Veneziano and E. Viada
Journal: Trans. Amer. Math. Soc. 369 (2017), 6465-6491
MSC (2010): Primary 11G50; Secondary 14G40
Published electronically: March 6, 2017
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Abstract: The Torsion Anomalous Conjecture states that an irreducible variety $ V$ embedded in a semi-abelian variety contains only finitely many maximal $ V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all maximal $ V$-torsion anomalous points of relative codimension one in the non-CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the Néron-Tate height of the rational points of an explicit family of curves of increasing genus.

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Additional Information

S. Checcoli
Affiliation: Institut Fourier, 100 rue des Maths, BP74 38402 Saint-Martin-d’Hères Cedex, France

F. Veneziano
Affiliation: Mathematisches Institut, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland

E. Viada
Affiliation: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany
Address at time of publication: ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland

Received by editor(s): July 2, 2013
Received by editor(s) in revised form: September 29, 2015
Published electronically: March 6, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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