Abstract
Let be an elliptic curve. An irreducible algebraic curve embedded in is called weak-transverse if it is not contained in any proper algebraic subgroup of , and transverse if it is not contained in any translate of such a subgroup.
Suppose and are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve that are close to the union of all algebraic subgroups of of codimension translated by points in a subgroup of of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If is trivial, it is sufficient to suppose that is weak-transverse.
The core of the article is the introduction of a method to determine the finiteness of these sets. From a conjectural lower bound for the normalized height of a transverse curve , we deduce that the sets above are finite. Such a lower bound exists for .
Concerning the codimension of the algebraic subgroups, our results are best possible.
Citation
Evelina Viada. "The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve." Algebra Number Theory 2 (3) 249 - 298, 2008. https://doi.org/10.2140/ant.2008.2.249
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