Introduction to Modern Number Theory

Fundamental Problems, Ideas and Theories

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 49)

Table of contents

  1. Front Matter
    Pages I-6
  2. Problems and Tricks

    1. Front Matter
      Pages 7-7
    2. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 9-61
    3. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 63-91
  3. Ideas and Theories

    1. Front Matter
      Pages 93-93
    2. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 95-114
    3. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 115-189
    4. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 191-259
    5. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 261-340
    6. Yuri Ivanovic Manin, Alexei A. Panchishkin
      Pages 341-393
  4. Analogies and Visions

  5. Back Matter
    Pages 461-514

About this book

Introduction

"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)

Keywords

Arithmetic Arithmetic der algebraischen Zahlen Elementare Zahlentheorie Elementary number theory Langlands program Langlands-Programm algebraic varieties arithmetic of algebraic numbers commutative property diophantine equations diophantische Gleichungen elliptic curves elliptische Kurven number theory public

Authors and affiliations

  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Institut FourierUniversité Joseph Fourier UMR 5582Saint Martin d’HèresFrance

Bibliographic information

  • DOI https://doi.org/10.1007/3-540-27692-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 2005
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-20364-3
  • Online ISBN 978-3-540-27692-0
  • Series Print ISSN 0938-0396
  • About this book