Capacity theory on algebraic curves
Author(s)
Bibliographic Information
Capacity theory on algebraic curves
（Lecture notes in mathematics, 1378）
Springer-Verlag, c1989
- : gw
- : us
Available at 74 libraries
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Library & Science Information Center, Osaka Prefecture University
: gwNDC8:410.8||||10009557163
Note
Bibliography: p. [423]-428
Includes index
Description and Table of Contents
Description
Capacity is a measure of size for sets, with diverse applications in potential theory, probability and number theory. This book lays foundations for a theory of capacity for adelic sets on algebraic curves. Its main result is an arithmetic one, a generalization of a theorem of Fekete and Szegoe which gives a sharp existence/finiteness criterion for algebraic points whose conjugates lie near a specified set on a curve. The book brings out a deep connection between the classical Green's functions of analysis and Neron's local height pairings; it also points to an interpretation of capacity as a kind of intersection index in the framework of Arakelov Theory. It is a research monograph and will primarily be of interest to number theorists and algebraic geometers; because of applications of the theory, it may also be of interest to logicians. The theory presented generalizes one due to David Cantor for the projective line. As with most adelic theories, it has a local and a global part. Let /K be a smooth, complete curve over a global field; let Kv denote the algebraic closure of any completion of K. The book first develops capacity theory over local fields, defining analogues of the classical logarithmic capacity and Green's functions for sets in (Kv). It then develops a global theory, defining the capacity of a galois-stable set in (Kv) relative to an effictive global algebraic divisor. The main technical result is the construction of global algebraic functions whose logarithms closely approximate Green's functions at all places of K. These functions are used in proving the generalized Fekete-Szegoe theorem; because of their mapping properties, they may be expected to have other applications as well.
Table of Contents
Preliminaries.- Foundations.- The canonical distance.- Local capacity theory - Archimedean case.- Local capacity theory - Nonarchimedean case.- Global capacity theory.- Applications.
by "Nielsen BookData"