# zbMATH — the first resource for mathematics

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
Full Text: