Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heights in families of abelian varieties and the Geometric Bogomolov
  Conjecture   On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a
symmetric relatively ample line bundle defines a fiberwise N\'eon-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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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas