
Introduction  Preview Material  Table of Contents  Index  Supplementary Material 
Mathematical Surveys and Monographs 2013; 437 pp; hardcover Volume: 193 ISBN10: 1470409801 ISBN13: 9781470409807 List Price: US$119 Member Price: US$95.20 Order Code: SURV/193 See also: Potential Theory and Dynamics on the Berkovich Projective Line  Matthew Baker and Robert Rumely An Invitation to Arithmetic Geometry  Dino Lorenzini  This book is devoted to the proof of a deep theorem in arithmetic geometry, the FeketeSzegö 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 \([a,b]\) 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 nearextremal approximating functions by means of the canonical distance. Readership Graduate students and research mathematicians interested in arithmetic geometry, number theory, potential theory, algebraic geometry, and dynamics. 


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