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An explicit Manin-Dem'janenko theorem in elliptic curves

  • Evelina Viada,
    Mathematisches Institut, Georg-August-Universität, Bunsenstraße 3-5, D-D-37073, Göttingen, Germany
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Abstract

Let $\mathcal{C}$ be a curve of genus at least $2$ embedded in $E_1 \times \cdots \times E_N$ where the $E_i$ are elliptic curves for $i=1,\dots, N$. In this article we give an explicit sharp bound for the Néron-Tate height of the points of $\mathcal{C}$ contained in the union of all algebraic subgroups of dimension $\lt \max(r_\mathcal{C}-t_\mathcal{C},t_\mathcal{C})$ where $t_\mathcal{C}$, respectively $r_\mathcal{C}$, is the minimal dimension of a translate, respectively of a torsion variety, containing $\mathcal{C}$. 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 in products of elliptic curves.
Keywords: height, elliptic curve, explicit Mordell conjecture, explicit Manin-Demjanenko theorem, rational points on a curve height, elliptic curve, explicit Mordell conjecture, explicit Manin-Demjanenko theorem, rational points on a curve
MSC Classifications: 11G50, 14G40 show english descriptions Heights [See also 14G40, 37P30]
Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
11G50 - Heights [See also 14G40, 37P30]
14G40 - Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
 

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