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LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES
Journal article

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

  • VIADA, EVELINA Department of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
  • 2011-11-21
Published in:
  • International Journal of Number Theory. - World Scientific Pub Co Pte Lt. - 2010, vol. 06, no. 03, p. 471-499
Algebra and Number Theory
English This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d 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. Then, 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.
Language
  • English
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closed
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https://sonar.ch/global/documents/182180
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