Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007 (Lecture Notes in Mathematics (2009)) (English and French Edition)

Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties over arbitrary rings, in particular over non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thélène Peter Swinnerton Dyer and Paul Vojta. Annotation Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta Variétés Presque Rationnelles, Leurs Points Rationnels Et Leurs Dégénérescences / Jean-louis Colliot-thélène -- Topics In Diophantine Equations / Peter Swinnerton-dyer -- Diophantine Approximation And Nevanlinna Theory / Paul Vojta. Jean Louis Colliot Thélène, Peter Swinnerton-dyer, Paul Vojta ; Editors, Pietro Corvaja, Carlo Gasbarri. Includes Bibliographical References And Index. Text In English And French. Front Matter....Pages i-xi Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences....Pages 1-44 Topics in Diophantine Equations....Pages 45-110 Diophantine Approximation and Nevanlinna Theory....Pages 111-224 Back Matter....Pages 225-232 At the intersection of classical algebraic geometry and number theory, arithmetic geometry studies algebraic varieties through arbitrary rings. This volume collates written-up lecture notes from the 2007 ACIME summer school and covers numerous relevant topics.