Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

Abstract

On an abelian scheme over a smooth curve over $\overline {\mathbb {Q}}$ a symmetric relatively ample line bundle defines a fiberwise Néron–Tate height. If the base curve is inside a projective space, we also have a height on its $\overline {\mathbb {Q}}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\overline {\mathbb {Q}}$. Using Moriwaki’s height we sketch how to extend our result when the base field of the curve has characteristic $0$.

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      pages = {xvi+496},
      isbn = {0-387-90244-9},
      mrclass = {14-01},
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      }
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      author = {Hindry, Marc},
      title = {Autour d'une conjecture de {S}erge {L}ang},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
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      number = {3},
      pages = {575--603},
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      mrclass = {11G10 (11J89 14K15)},
      mrnumber = {0969244},
      mrreviewer = {Michel Laurent},
      doi = {10.1007/BF01394276},
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      title = {Diophantine Geometry. An Introduction},
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      mrclass = {11Gxx (11-02 11G10 11G30 11G50 14G25)},
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      doi = {10.1007/978-1-4612-1210-2},
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      mrnumber = {1854232},
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      doi = {10.1016/S0168-0072(01)00096-3},
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      mrclass = {11-02 (11Dxx 11Gxx 14G25)},
      mrnumber = {0715605},
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      doi = {10.1007/978-1-4757-1810-2},
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      journal = {Amer. J. Math.},
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      doi = {10.2307/2372851},
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      year = {1983},
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Authors

Ziyang Gao

CNRS, IMJ-PRG, Paris, France; Department of Mathematics, Princeton University, Princeton, NJ USA

Current address:

Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät, für Mathematik und Physik, Leibniz Universität Hannover, 30167 Hannover, Germany Philipp Habegger

Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland