Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49)

''Introduction to Modern Number Theory'' surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects. From the reviews of the 2nd edition: ''… For my part, I come to praise this fine volume. This book is a highly instructive read … the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly up-to-date ...'' (A. van der Poorten, Gazette, Australian Math. Soc. 34 (1), 2007) This edition has been called startlingly up-to-date, and in this corrected second printing you can be sure that its even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Front Matter....Pages 7-7 Number Theory....Pages 9-61 Some Applications of Elementary Number Theory....Pages 63-91 Front Matter....Pages 93-93 Induction and Recursion....Pages 95-114 Arithmetic of algebraic numbers....Pages 115-189 Arithmetic of algebraic varieties....Pages 191-259 Zeta Functions and Modular Forms....Pages 261-340 Fermat’s Last Theorem and Families of Modular Forms....Pages 341-393 Front Matter....Pages 393-393 Introductory survey to part III: motivations and description....Pages 397-413 Arakelov Geometry and Noncommutative Geometry (d’après C. Consani and M. Marcolli, [CM])....Pages 415-460 Arakelov Geometry and Noncommutative Geometry (d’après C. Consani and M. Marcolli, [CM])....Pages 415-460