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      Heights of points with bounded ramification

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          Abstract

          Let \(E\) be an elliptic curve defined over a number field \(K\) with fixed non-archimedean absolute value \(v\) of split-multiplicative reduction, and let \(f\) be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on \(E\) is either zero or bounded from below by a positive constant, for all points of bounded ramification over \(v\). In this paper we make this bound effective and prove an analogue result for the canonical height associated to \(f\). We also study variations of this result by changing the reduction type of \(E\) at \(v\). This will lead to examples of fields \(F\) such that the N\'eron-Tate height on non-torsion points in \(E(F)\) is bounded from below by a positive constant and the height associated to \(f\) gets arbitrarily small on \(F\). The same example shows, that the existence of such a lower bound for the N\'eron-Tate height is in general not preserved under finite field extensions.

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          Sinc approximation of algebraically decaying functions

          An extension of sinc interpolation on \(\mathbb{R}\) to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of error estimates. First covers a wider class of functions with the algebraic order of decay on \(\mathbb{R}\). The second type of error estimates governs the case when the order of function's decay can be estimated everywhere in the horizontal strip of complex plane around \(\mathbb{R}\). The numerical examples are provided.
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            Advanced Topics in the Arithmetic of Elliptic Curves

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              The Bogomolov conjecture for totally degenerate abelian varieties

              We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
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                Author and article information

                Journal
                16 January 2012
                2013-11-18
                Article
                1201.3327
                00c5c099-6e4f-4af4-9b9b-c08250521288

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                11G50 (Primary) 37P30, 14H52 (Secondary)
                Major changes regarding the first version. E.g.: the title was changed; errors were corrected; based on a remark of Joseph Silverman Example 5.7 and Theorem 5.9 were added
                math.NT

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