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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 SzegA 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 NA(c)ron'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-SzegA theorem; because of their mapping properties, they may
be expected to have otherapplications as well.
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