Nondense Subsets of Varieties in a Power of an Elliptic Curve

Abstract

Let E be an elliptic curve without complex multiplication (CM) defined over Q. We show that on a transverse d -dimensional variety V in a g-power of E, the set of algebraic points of bounded height, which are close to the union of all algebraic subgroups of E g of codimension d + 1 translated by points in a subgroup of finite rank, is Zariski nondense in V. The notion of close is defined using a height function. If the subgroup of finite rank is trivial, it is sufficient to assume that V is weak-transverse. This result is optimal with respect to the codimension of the algebraic subgroups. The method is based on an essentially optimal effective version of the Bogomolov Conjecture. Such an effective result is proven for subvarieties of a product of ellipitc curves. If we assume that the sets have bounded height, then we can prove that they are not Zariski dense. A conjecture, known in some special cases, claims that the sets in question have bounded height. We prove here a new case. In conclusion, our results prove a generalized case of a conjecture by Zilber and by Pink in a power of E.