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An explicit ManinDem'janenko theorem in elliptic curves


Evelina Viada,
Mathematisches Institut, GeorgAugustUniversität, Bunsenstraße 35, DD37073, Göttingen, Germany
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 NéronTate 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 ManinDem'janenko
method in products of elliptic curves.