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An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

Evelina Viada
Evelina Viada*
Affiliation:
Mathematisches Institut, Georg-August-Universitä, Bunsenstraße 3-5, D-D-37073, Göttingen, Germany, e-mail: evelina.viada@mathematik.uni-goettingen.de
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Abstract

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Let $c$ be a curve of genus at least 2 embedded in ${{E}_{1}}\,\times \,.\,.\,.\,\times \,{{E}_{N}}$ , where the ${{\text{E}}_{i}}$ are elliptic curves for $i\,=\,1,\,.\,.\,.\,,\,N$. In this article we give an explicit sharp bound for the Néron–Tate height of the points of $c$ contained in the union of all algebraic subgroups of dimension $<\,\max ({{r}_{C}}\,-{{t}_{C}},\,{{t}_{C}})$ , where ${{t}_{C}}(\text{resp}\text{.}{{r}_{C}})$ is the minimal dimension of a translate (resp. of a torsion variety) containing $c$.

As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.

Keywords

11G5014G40heightelliptic curveexplicit Mordell conjectureexplicit Manin-Demjanenko theoremrational points on a curve
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 1173 - 1200
Copyright
Copyright © Canadian Mathematical Society 2018

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