"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)

Table of Contents

Part I Problems and Tricks

1 Elementary Number Theory

2 Some Applications of Elementary Number Theory

Part II Ideas and Theories

3 Induction and Recursion

4 Arithmetic of algebraic numbers

5 Arithmetic of algebraic varieties

6 Zeta Functions and Modular Forms

7 Fermat’s Last Theorem and Families of Modular Forms

Part III Analogies and Visions

8 Arakelov Geometry and Noncommutative Geometry