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Arithmetics. Primality and codes, analytic number theory, Diophantine equations, elliptic curves. (Arithmétique. Primalité et codes, théorie analytique des nombres, équations diophantiennes, courbes elliptiques.) (French) Zbl 1135.11001

Tableau Noir 102. Paris: Calvage et Mounet (ISBN 978-2-916352-04-6/hbk). xvi, 328 p. (2008).
The book under review offers a basic course in number theory of a very special kind. Geared toward graduate students at the masters level (M1 and M2), this text provides a thorough and lively introduction to various fundamental aspects of both classical and contemporary arithmetical theories, together with some of their most important applications and current research developments. Instead of focussing primarily on one of the many different branches in modern number theory (like analytic number theory, algebraic number theory, transcendental number theory, arithmetic geometry, and several others), as it is common for most textbooks in this field, the present book emphasizes an utmost enlightening, inspiring, multifarious and fascinating panoramic approach to number theory as a whole, thereby making transparent, in a likewise manner, all the great features of this venerable mathematical discipline, which C. F. Gauss once declared the “queen of mathematics”. In fact, only a broad panoramic view to number theory can reveal all these outstanding characteristics in their entirety, above all its unique genesis, its intrinsic beauty, its ubiquity in contemporary pure and applied mathematics, its methodological variety, its refined mathematical spirituality, and its everlasting, permanently rejuvenating topicality in mathematics.
In this vein, the author of the current book has aligned both his didactic and methodological principles in expository writing so as to convey such a representative insight into the diversity of the methods of number theory, and the outcome is a unique and masterly primer of advanced arithmetic.
As for the contents, the text is composed of six Chapters and three Appendices, each of which is subdivided into several sections.
The first part of the book encompasses Chapters I–IV and contains the material of a masters course in arithmetic as it is usually taught at Université de Paris 7 and at École Normale Supérieure in Paris. This part is throughout very detailed, and the proofs of all results discussed here are given in entire completeness.
On the other hand, the second part (Chapters V, VI and the Appendices) appears to be more advanced, explanatory, survey-like and already quite close to the forefront of current research in the field. This part is rather designed as both an incentive and a source for possible masters theses and further research work in contemporary arithmetic, which must be seen as another distinctive feature of the book under review.
More precisely, the actual contents of the respective chapters are organized as follows:
Chapter I provides a systematic introduction to the structure of the finite groups and rings \(\mathbb Z/n\mathbb Z\), the finite fields \(\mathbb F_q\), and their groups of units. This, together with the classical facts on Gaussian sums, is then applied to derive a proof of Gauss’s quadratic law of reciprocity, on the one hand, and to the study of the solutions of certain algebraic equations over finite fields on the other.
After these classical topics, Chapter II turns immediately to the more recent practical applications of number theory to the fields of informatics, cryptography, and coding theory. This includes the basic algorithms in number theory and their complexity, the principles of RSA cryptography, the problem of primality and various primality tests, factorization procedures for integers, and a brief account of linear cyclic codes (Hamming codes, Reed-Solomon codes, Golay codes, and others). The necessary facts about cyclotomic polynomials are developed along the way.
Chapter III returns to classical number theory and is devoted to an introduction to the magic realm of Diophantine equations. The author discusses the problem of representing positive integers as sums of squares, the famous Fermat equation for the exponents 3 and 4, the important Pell-Fermat equation \(x^2- dy^2=1\) and, along this path, the allied modern concepts of the algebraic theory of numbers. The latter incorporates the fundamentals of algebraic number fields, rings of algebraic integers and their (prime) ideal theory, groups of units, the finiteness of the ideal class group of a number field, and the related toolkit from geometric number theory (Minkowski’s theorem on lattice points) and Diophantine approximation (continued fractions, Dirichlet’s finiteness theorem on groups of units).
Finally, in order to complete the first panoramic view to basic number theory, Chapter IV gives an introduction to some central aspects of analytic number theory. The leading theme of this chapter is the distribution of prime numbers, culminating in detailed proofs of the celebrated prime number theorem and of Dirichlet’s theorem on primes in arithmetic progressions. In this context, Dirichlet series, the Riemann zeta-function, and \(L\)-series are thoroughly discussed as well, together with an outlook to the famous Riemann hypothesis and further related topics of current research.
The second, more advanced and topical part of the book begins with Chapter V, in which the study of elliptic curves is initiated. The author depicts the various algebraic, analytic and arithmetic properties of these fascinating objects in a very concise and elegant manner, thereby illustrating the marvelous interplay of different mathematical theories and methods. However, the main objective of this chapter is to demonstrate the arithmetic significance of elliptic curves, and this is done by means of two of the great theorems in Diophantine geometry: the finiteness theorem of Mordell-Weil for the group of rational points on an arithmetic elliptic curve, on the one hand, and the finiteness theorem of Siegel for integer points on the other. In the course of the discussion, the concept of height functions (à la Weil and Néron-Tate) is developed, and a beautiful outlook to the recent work of A. Wiles on modular elliptic curves and Fermat’s Last Theorem as well as to the related famous conjecture of Birch and Swinnerton-Dyer is given at the end of this chapter.
The link to recent developments and still open problems in number theory is pursued in the final Chapter VI, which is much more advanced, sketchy and related to current research than the others. Although being far beyond the scope of a basic course in arithmetic, the modern topics presented here not only give the budding arithmetician an overwhelming glimpse of the trends and prospects in contemporary number theory, but also may serve as a perfect guide to individual research activities in various directions, and as a great source for additional, extended reading likewise. To this end, the author has chosen the following six themes reflecting some of the most important and spectacular advances in number theory achieved over the past decades:
(1) Algebraic varieties over finite fields, their zeta-functions, the Weil conjectures, and the related results by A. Weil, S. Lang, A. Grothendieck, P. Deligne, and others.
(2) Algebro-geometric aspects of Diophantine equations, the famous Lang conjectures, the Mordell conjecture and the finiteness theorem of Faltings, the link to analytic hyperbolicity properties of complex varieties, and Siegel’s finiteness theorem for integer points on arithmetic curves.
(3) Completions, \(p\)-adic numbers, Hensel’s lemma, the Hasse principle and a brief introduction to adeles and ideles.
(4) Transcendental numbers and Diophantine approximation in the light of the fundamental results by Thue, Siegel, Roth, and Baker.
(5) The abc-conjecture (à la Masser-Oesterlé) and its relations to the arithmetic of elliptic curves, the conjectures of Frey and Szpiro, Belyi’s theory, and the great theorem of Fermat-Wiles.
(6) Generalizations of Dirichlet series, modular forms, Hecke operators and \(L\)-functions, Galois representations, related results by Serre and Deligne, a reformulation of Wiles’s theorem with a view to the Shimura-Taniyama-Weil conjecture, and an explanation of the Hasse-Weil conjecture.
Each section refers to several related open problems, and there is plentiful supply of hints for further, more detailed reading. Also, the reader finds here many additional comments pointing at other related current research areas such as the celebrated Langlands programme, Grothendieck’s theory of motives, and others.
In order to keep the entire text as self-contained as possible, with only the basics of algebra and real analysis as assumed background knowledge, the author has concisely recalled the more advanced prerequisites from complex analysis, projective algebraic geometry, factorization theory in special rings, and higher Galois theory in an appropriate manner, either within the text itself or in the three appendices. Appendix A provides the necessary factorization methods and algorithms for arithmetical rings and elliptic curves, whereas Appendix B introduces a few fundamental concepts from projective algebraic geometry. Finally, Appendix C compiles the basic facts from Galois theory of number fields, abelian field extensions, and Galois representations as they are used in Chapter VI. For the convenience of the reader, there is a rich (and partly commented) bibliography of about 80 references at the end of the book, accompanied by a very detailed index and an utmost careful list of notations.
Altogether, the book under review is both, a brilliant introduction and a truly irresistible invitation to the magic world of number theory in all its fascinating aspects. Written in a masterly lucid, didactically refined and mathematically utmost profound style, this book is a masterpiece of expository writing in mathematics. It covers a vast and opalescent spectrum of central topics in both classical and contemporary number theory, with hundreds of carefully selected exercises woven into the main text, and as such it provides an invaluable source book for professors, instructors, and young researchers in the field, too. This enchanting panorama of arithmetic has certainly got what it takes to become a standard introduction to the subject, a widely popular textbook, and a bestseller besides. Such a great textbook at an affordable price – that has become very rare! No doubt, the author and the editor have done a great favour to the mathematical community, and therefore it remains to be wished that this excellent textbook will find a worldwide audience of readers through a translation into English – the sooner the better!

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11G05 Elliptic curves over global fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94Bxx Theory of error-correcting codes and error-detecting codes
11Mxx Zeta and \(L\)-functions: analytic theory
11Dxx Diophantine equations
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11A51 Factorization; primality
11Yxx Computational number theory
11Jxx Diophantine approximation, transcendental number theory
11Txx Finite fields and commutative rings (number-theoretic aspects)

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