Small points on constant arithmetic surfaces
type: Item
instance of: scholarly article
title: Small points on constant arithmetic surfaces
zbMATH Open document ID: 0727.14016
DOI: 10.1215/S0012-7094-90-06131-9
author: Minhyong Kim
published in: Duke Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The paper under review considers Arakelov theory. That is we start with a regular semistable curve X over the integers of a number-field, and can define an intersection theory for Arakelov divisors on X very much as for algebraic surfaces. Such a surface, of genus \(g\geq 2,\) is called constant if its dualising sheaf \(\omega_ X\) satisfies \(\omega^ 2_ X=0\). It is known that always \(\omega^ 2_ X\geq 0\), and there is some evidence for the conjecture that constant arithmetic surfaces do not exist. Anyway in the present paper the author first constructs sections of suitably bounded norm in \(\Gamma (X,\omega^ n)\), and then uses them to construct an infinite sequence of algebraic points whose Néron-Tate heights converge to zero.
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 4200374
zbMATH Keywords: small points
zbMATH Keywords: constant arithmetic surfaces
zbMATH Keywords: Arakelov theory
zbMATH Keywords: intersection theory
zbMATH Keywords: Néron-Tate heights
MaRDI profile type: Publication
cites work: An Introduction to the Geometry of Numbers
cites work: Q3765893
cites work: Calculus on arithmetic surfaces
cites work: Arithmetic intersection theory
cites work: Characteristic classes for algebraic vector bundles with Hermitian metric. I
cites work: Q3831122
cites work: Numerically positive line bundles on an Arakelov variety
cites work: Q3994900
cites work: Courbes sur une variété abélienne et points de torsion
full work available at URL: s0012-7094-90-06131-9
OpenAlex ID: W1979202831
reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 4200374