Capacity Theory with Local Rationality: The Strong Fekete-Szegö Theorem on Curves
About this ebook
theorem in arithmetic geometry, the Fekete-Szegö theorem with local
rationality conditions. The prototype for the theorem is Raphael
Robinson's theorem on totally real algebraic integers in an interval,
which says that if
is a real interval of length greater than 4, then it contains
infinitely many Galois orbits of algebraic integers, while if its
length is less than 4, it contains only finitely many. The theorem
shows this phenomenon holds on algebraic curves of arbitrary genus
over global fields of any characteristic, and is valid for a broad
class of sets.
The book is a sequel to the author's work Capacity Theory on Algebraic Curves
and contains applications to algebraic integers and units, the
Mandelbrot set, elliptic curves, Fermat curves, and modular curves. A
long chapter is devoted to examples, including methods for computing
capacities. Another chapter contains extensions of the theorem,
including variants on Berkovich curves.
The proof uses both
algebraic and analytic methods, and draws on arithmetic and algebraic
geometry, potential theory, and approximation theory. It introduces new
ideas and tools which may be useful in other settings, including the
local action of the Jacobian on a curve, the "universal function" of
given degree on a curve, the theory of inner capacities and Green's
functions, and the construction of near-extremal approximating
functions by means of the canonical distance.
