Lower bounds for the normalized height and non-dense subsets of varieties in an abelian var.pdfVIP

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Lower bounds for the normalized height and non-dense subsets of varieties in an abelian var.pdf

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Lower bounds for the normalized height and non-dense subsets of varieties in an abelian var

a r X i v : 0 8 0 6 .0 4 8 7 v 1 [ m a t h .N T ] 3 J u n 2 0 0 8 LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF VARIETIES IN AN ABELIAN VARIETY Evelina Viada1 2 3 Abstract. This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak- transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V . For V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank are non Zariski- dense in V . If Γ has rank zero, it is sufficient to assume that V is weak- transverse. The notion of closeness is defined using a height function. 1. introduction All varieties in this article are defined over Q. Denote by A a abelian variety of dimension g. Consider an irreducible algebraic subvariety V of A of dimension d. We say that ? V is transverse, if V is not contained in any translate of a proper algebraic subgroup of A. ? V is weak-transverse, if V is not contained in any proper algebraic subgroup of A. As we are going to consider only algebraic points, we denote by A = A(Q) and V = V (Q). For a subset S of A, we denote by S its Zariski closure. Given a subset V e of V , an integer k with 1 ≤ k ≤ g and a subset F of A, we define the set (1) Sk(V e, F ) = V e ∩ ? codB≥k B + F, where B varies over all abelian subvarieties of A of codimension at least k and B + F = {b+ f : b ∈ B, f ∈ F}. We denote the set Sk(V e, ATor) simply by Sk(V e), where ATor is the torsion of A. Nowadays a v