On some applications of diophantine approximations [electronic resource] : a translation of C.L. Siegel’s Über einige Anwendungen diophantischer Approximationen, with a commentary by C. Fuchs and U. Zannier) /
Zannier, Umberto.editor.
This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.
Cover; Title Page; Copyright Page; Table of Contents; Preface; On some applications of Diophantine approximations; 1 Part I: On transcendental numbers; 1 Tools from complex analysis; 2 Tools from arithmetic; 3 The transcendence of J0(ξ); 4 Further applications of the method; 2 Part II: On Diophantine equations; 1 Equations of genus 0; 2 Ideals in function fields and number fields; 3 Equations of genus 1; 4 Auxiliary means from the theory of ABEL functions; 5 Equations of arbitrary positive genus; 6 An application of the approximation method; 7 Cubic forms with positive discriminant
Über einige Anwendungen diophantischer ApproximationenIntegral points on curves: Siegel's theorem after Siegel's proof; 1 Introduction; 2 Some developments after Siegel's proof; 3 Siegel's Theorem and some preliminaries; 4 Three arguments for Siegel's Theorem; References; MONOGRAPHS
With a commentary and the article Integral points on curves: Siegel's theorem after Siegel's proof by Clemens Fuchs and Umberto Zannier.
Includes bibliographical references.
This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.
Number theory.
Number Theory.