Capacity theory on algebraic curves
type: Item
instance of: scholarly article
title: Capacity theory on algebraic curves
zbMATH Open document ID: 0679.14012
author: Robert S. Rumely
published in: Lecture Notes in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: In the theory of functions of one complex variable, capacity is a measure of size for compact sets. This concept of capacity plays an essential role in the study of harmonic functions, equilibrium distribution, and general potential theory within the theory of one complex variable. On the other hand, it provides significant applications to algebraic number theory, too. By a classical theorem of \textit{M. Fekete} and \textit{G. Szegö} [cf. Math. Z. 63, 158--172 (1955; Zbl 0066.27002)], the distribution of complete conjugate sets of algebraic integers with respect to a compact set in the complex plane can be measured by the logarithmic capacity of such a set. The naturally arising question of whether there is an analogous theorem for arbitrary number fields (rather than \({\mathbb{Q}})\) and more general sets at the finite primes has been partially answered by D. Cantor (1967, 1980) who defined a generalized capacity measure for adelic sets in the projective line, and proved, together with P. Roquette in 1984, a generalization of the Fekete-Szegö theorem, and a general existence theorem on rational points in affine varieties defined over arbitrary number fields. In the present monograph, the author now extends capacity theory to arbitrary curves (of any genus) with an arbitrary global field as ground field. This pretentious enterprise requires to lay fairly new foundations, as on curves of higher genus (instead of \({\mathbb{P}}^ 1)\) one can only lean upon local methods. The author's far-reaching work to found a generalized capacity theory for algebraic curves and to apply it to arithmetical problems is based upon an ingenious combination of the classical methods, on the one hand, and several essentially new ingredients, such as local parametrizations by power series, a certain global maximum modulus principle, an existence theorem for algebraic functions with near-prescribed divisors, the Deligne-Mumford theorem on stable reduction, and Néron's canonical local heig…
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 30-02
zbMATH DE Number: 43463
zbMATH Keywords: capacity
zbMATH Keywords: distribution of complete conjugate sets of algebraic integers
zbMATH Keywords: local parametizations by power series
zbMATH Keywords: global maximum modulus principle
zbMATH Keywords: stable reduction
zbMATH Keywords: local height pairing on curves
zbMATH Keywords: canonical distance functions on algebraic curves
zbMATH Keywords: capacity measures
zbMATH Keywords: Green functions
reviewed by: Werner Kleinert
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0084525
OpenAlex ID: W1560501720
DOI: 10.1007/BFB0084525
description: scientific article; zbMATH DE number 43463