An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

Viada, Evelina

Canadian journal of mathematics, Tome 70 (2018) no. 5, pp. 1173-1200

Résumé

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.

Détail

DOI : 10.4153/CJM-2017-045-0

Mots-clés : 11G50, 14G40, height, elliptic curve, explicit Mordell conjecture, explicit Manin-Demjanenko theorem, rational points on a curve