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Zbl 0852.14005
Boxall, John
A property of local Néron-Tate heights on abelian varieties. (Une propriété des hauteurs locales de Néron-Tate sur les variétés abéliennes.)
(French)
[J] J. Théor. Nombres Bordx. 7, No.1, 111-119 (1995). ISSN 1246-7405

Suppose $K$ is the completion of a number field at a finite prime and $E/K$ is an elliptic curve with good reduction. If $h$ is the local height function on $E$ corresponding to the divisor $D= (0)$, then explicit formulas of Tate show that $$\lim_{N\to \infty} {1\over {N^2}} \sum_{0\ne P\in E[ N]} h(P) =0.$$ The limit on the left hand side arises in the computation of normalizations of local heights. \par The author proves a version of the equality above that is valid for abelian varieties of arbitrary dimension. In particular, he shows that if $A$ is a $g$-dimensional abelian variety over $K$ with good reduction, if $h$ is the local height function on $A$ corresponding to a divisor $D$, and if $p$ is the residue characteristic of $K$, then $$\lim\Sb N\to \infty\\ N\not\equiv 0\bmod p\endSb {1\over {N^{2g}}} \sum\Sb P\in A[ N]\\ P\not\in \text {Supp } D\endSb h(P) =0.$$ The proof of this equality splits into two parts. In the first, the author shows that the number of non-zero terms in each sum is $O( N^{2g- 2})$. In the second, he uses an argument involving the Greenberg functor [see {\it M. J. Greenberg}, Ann. of Math., II. Ser. 73, 624-628 (1961; Zbl 0115.39004)] to show that the size of these terms is bounded above by a constant independent of $N$. The proof of the theorem is preceded by an introduction that reviews the normalization of local height functions for elliptic curves and by a section that lists the properties of the Greenberg functor used in the paper.
[W.E.Everett (San Diego)]
MSC 2000:
14K05 Algebraic theory of abelian varieties
11G10 Abelian varieties of dimension $>1$

Keywords: local height function; abelian varieties

Citations: Zbl 0115.39004

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