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, 4 place de Jussieu, 75005 Paris, France; Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

Philipp Habegger

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