Abstract
Let E be an elliptic curve. An irreducible algebraic curve C embedded in a power A of E is called weak-transverse if it is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Suppose E and C are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve C that are close to the union of all algebraic subgroups of E g of codimension 2 translated by points in a subgroup G of A of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If G is trivial, it is sufficient to suppose that C is weak-transverse. The core of the article is the introduction of a method to determine the finite- ness of these sets. From a conjectural lower bound for the normalized height of a transverse curve C , we deduce that the sets above are finite. Such a lower bound exists for g ≤ 3. Concerning the codimension of the algebraic subgroups, our results are best possible.
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
