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The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve. (English) Zbl 1168.11024
In connexion with the conjectures of Mordell-Lang, Manin-Mumford and Bogomolov, several authors investigated the intersection of the set of algebraic points over the field \(\bar{\mathbb Q}\) of algebraic numbers on a subvariety \(V\) of a semi-abelian variety \(A\) on the one hand, with the union of translates of semi-abelian subvarieties of \(A\) on the other hand. Here, the author mainly considers the case where, firstly, \(A=E^g\), with \(E\) an elliptic curve defined over \(\bar{\mathbb Q}\), secondly, \(V=C\) is an irreducible algebraic curve also defined over \(\bar{\mathbb Q}\), and thirdly, the semi-abelian subvarieties (here the subgroups of \(E^g\)) have codimension \(2\). She also investigates the more general situation of points which are close to such intersections, where the notion of closeness involves a height function. She introduces a new efficient way to show the finiteness of such sets. Further, she shows that a conjectural lower bound for the normalized height of a transverse curve implies the finiteness of such sets. Furthermore, she reaches unconditional results for \(g\leq 3\).

11G05 Elliptic curves over global fields
11D45 Counting solutions of Diophantine equations
11G50 Heights
14K12 Subvarieties of abelian varieties
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