Manin I., Panchishkin A. - Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories (2nd ed.).pdf - Number Theory - jacpr

Encyclopaedia of Mathematical Sciences

Vo l ume 4 9

Number Theory I

Yuri Ivanovic Manin

Alexei A. Panchishkin

Introduction to

Modern Number

Theory

Fundamental Problems, Ideas and Theories

Second Edition

123

Authors

Yuri Ivanovic Manin

Max-Planck-Institut für Mathematik

Vivatsgasse 7

53111 Bonn, Germany

e-mail: manin@mpim-bonn.mpg.de

Alexei A. Panchishkin

Universite Joseph Fourier UMR 5582

Institut Fourier

38402 Saint Martin d’Heres, France

e-mail: alexei.pantchichkine@ujf-grenoble.fr

Founding editor of the Encyclopaedia of Mathematical Sciences:

R. V. Gamkrelidze

Original Russian version of the ﬁrst edition

was published by VINITI, Moscow in 1990

The ﬁrst edition of this book was published as Number Theory I,

Yu. I. Manin, A. A. Panchishkin (Authors), A. N. Parshin, I. R. Shafarevich (Eds.),

Vol. 49 of the Encyclopaedia of Mathematical Sciences

Mathematics Subject Classif ication (2000):

11-XX (11A, 11B, 11D, 11E, 11F, 11G, 11R, 11S, 11U, 11Y), 14-XX, 20-XX, 37-XX, 03-XX

ISSN 0938-0396

ISBN-10 3-540-20364-8 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-20364-3 Springer Berlin Heidelberg New York

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Preface

The present book is a new revised and updated version of “Number Theory

I. Introduction to Number Theory” by Yu.I.Manin and A.A.Panchishkin, ap-

peared in 1989 in Moscow (VINITI Publishers) [Ma-PaM], and in English

translation [Ma-Pa] of 1995 (Springer Verlag).

The original book had been conceived as a part of a vast project, “En-

cyclopaedia of Mathematical Sciences”. Accordingly, our task was to provide

a series of introductory essays to various chapters of number theory, lead-

ing the reader from illuminating examples of number theoretic objects and

problems, through general notions and theories, developed gradually by many

researchers, to some of the highlights of modern mathematics and great, some-

times nebulous designs for future generations.

In preparing this new edition, we tried to keep this initial vision intact. We

present many precise deﬁnitions, but practically no complete proofs. We try

to show the logic of number-theoretic thought and the wide context in which

various constructions are made, but for detailed study of the relevant materials

the reader will have to turn to original papers or to other monographs. Because

of lack of competence and/or space, we had to - reluctantly - omit many

fascinating developments.

The new sections written for this edition, include a sketch of Wiles’ proof

of Fermat’s Last Theorem, and relevant techniques coming from a synthesis

of various theories of Part II; the whole Part III dedicated to arithmetical

cohomology and noncommutative geometry; a report on point counts on va-

rieties with many rational points; the recent polynomial time algorithm for

primality testing, and some others subjects.

For more detailed description of the content and suggestions for further

reading, see Introduction.

Manin I., Panchishkin A. - Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories (2nd ed.).pdf

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