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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 \(g\leq 3\).

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