Arithmetic distance functions and height functions in diophantine geometry
type: Item
instance of: scholarly article
title: Arithmetic distance functions and height functions in diophantine geometry
zbMATH Open document ID: 0607.14013
DOI: 10.1007/BF01461718
author: Joseph H. Silverman
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164317
review text: The classical theory of height functions associates to each (Cartier) divisor \(D\) on a projective variety \(V\) a function \(h_ D\) mapping the set of points of \(V\) to the real numbers. The function \(h_ D\) can be written as a sum of local height functions \(\lambda _ D(\cdot,v)\). (These local height functions are also known as logarithmic distance functions, Weil functions, or distributions). In this paper the author generalizes the theory of local height functions by associating a local height function \(\lambda _ X(\cdot,v)\) to each closed subscheme \(X\) of \(V\). This function gives a measure of the \(v\)-adic distance to \(X\). It has many nice functorial properties, such as \(\lambda _{X\cap Y}=\min \{\lambda _ X,\lambda _ Y\}\), which allow one to translate relations between closed subschemes into relations between local height functions. In this way, statements about geometry are easily converted into statements about arithmetic. Various applications are given, including a quantitative version of the inverse function theorem, and a description of the behavior of local height functions on proper families of varieties and on abelian schemes. More generally, the theory of local height functions developed in this paper should be useful whenever one has a parametric family of algebraic varieties, and is interested in studying how Diophantine properties vary in terms of the chosen point in the parameter space.
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 3981332
zbMATH Keywords: local height functions
zbMATH Keywords: closed subschemes
zbMATH Keywords: inverse function theorem
zbMATH Keywords: parametric family of algebraic varieties
MaRDI profile type: Publication
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Q4143433
cites work: Q3309989
description: scientific article; zbMATH DE number 3981332