Werner Kleinert (Berlin)**Faraut, Jacques**. Analysis on Lie Groups. An Introduction. (Analyse sur les groupes de Lie. Une introduction.) (French) [B] Paris : Calvage et Mounet. xi, 313 p. EUR 33.00 (2006). ISBN 2-916352-00-7/pbk

Altogether, this is an excellent introduction to the (harmonic) analysis on the classical linear Lie groups. Its didactic mastery is typified by the skillful choice of representative topics, by the resolute inductive style of teaching them, and by the instructive combination of geometric, algebraic and analytic methods from the modern point of view. This text comes with a distinctly perceivable, purposefully chosen structure, and it offers a wealth of concrete computations, examples, and applications. Together with the high degree of self-containedness, lucidity, fullness, and mathematical elegance, these features make the present text utmost user-friendly and recommendable. At any rate, this primer is a very welcome addition to the existing, more comprehensive and advanced standard texts on the subject, and an English translation of it would be more than just useful for the international mathematical community as a whole.

Werner Kleinert (Berlin)**Mneimné, Rached**. Reduction of Endomorphisms. Young Tableaux, Nilpotent Cone, Representations of Semi-simple Lie Algebras. (Réduction des endomorphismes. Tableaux de Young, cône nilpotent, représentations des algèbres de Lie semi-simples.) (French)

[B] Paris : Calvage et Mounet. xvii, 376 p. EUR 36.00 (2006). ISBN 2-916352-01-5/hbk

… the present book may be regarded as the first attempt in the textbook literature to focus solely on the Jordan normal form, and to analyze both its significance and its fascinating ubiquity in modern mathematics under various aspects. Moreover, the author has tried to exhibit the Jordan theory as what it virtually is : a cultural jewel of mathematics, overflowing with aesthetics, beauty, and infinite surprises, just like a vast field of marvellous flowers. This nearly belletristic approach, larded with numerous literary quotations, leads the reader from the elementary foundations of the Jordan reduction theorem up to its crucial role in combinatorics, in invariant theory, and in the theory of finite-dimensional complex Lie algebras, with an abundance of amazing extra topics that are mostly neglected in the contemporary textbook literature on these subjects…

It is fair to say that this panoramic textbook on the Jordan normal form is an aesthetic delicacy of a very special kind, and a great source of both motivation and inspiration besides. Although written for seasoned students, this book also offers a wealth of new insights even for experts in the field.

Werner Kleinert (Berlin)**Tauvel, Patrice**. Commutative Fields and Galois Theory. (Corps commutatifs et théorie de Galois.) (French)

[B] Mathématiques en Devenir. Paris : Calvage et Mounet. xiv, 348 p. EUR 35.00 (2007).

ISBN 978-2-916352-02-2

The textbook under review provides a very comprehensive, modern and elegant introduction to the theory of commutative fields in abstract algebra, together with a remarkable number of highly interesting novelties with regard to topical contents and methodological aspects.

In the course of the past two decades, an avalanche of introductory texts on abstract algebra, field theory, and Galois theory has virtually overstocked the international textbook market, with an increasing tenndency to keep the expostion as elementary, purposive, and beginner-friendly as possible, and to follow the historical path of development of the subject, embellished by Evariste Galois’s spectacular, touching biography as didactic instrument.

Instead of reiterating this widespread, unfortunately fashionable pattern in his present textbook, the author sets against it systematic depth, diversity, and conceptual generality. As he points out in the preface to his book, the material is deliberately presented on a more advanced level than in many other textbooks on the subject in order to exhibit the crucial significance of field theory in modern mathematics to an appropriate extent, on the one hand, and to counteract the recent flattening tendencies in the teaching programs at many universities on the other. This must be seen as a rewarding attempt to maintain the good old quality standards in teaching the subject within the actual reforms and changes, and to offer a constructive variant for both teachers and students. In this vein, the text is intended to be widely self-contained, therefore containing several chapters on related basic topics from group theory, polynomial algebra, number theory, and ring theory.

…

In view of the many outstanding features that this masterly textbook on fields and Galois theory has to offer, above all with regard to its methodological originality and elegance, mathematieal abundance and topicality, didactical skill, and user-friendly lucidity, it must be seen as both a highly valuable complement and a welcome alternative to the vast textbook literature in this fundamental area of modern abstract algebra. For the benefit of the international mathematical community, it would be more than desirable to make an English translation of this excellent textbook available.

Ilka Agricola (Berlin)**Saint Raymond, Jean.** Topology, Differential Calculus and Complex Variables. Course and Exercises. (Topologie, calcul différentiel et variable complexe. Cours et exercices.) (French)

[B] Mathématiques en Devenir. Paris : Calvage et Mounet. xiv, 442 p. EUR 39.00 (2007). ISBN 978-2-916352-03-9

This carefully written textbook consists of two parts, the first on elementary topology and functional analysis, the second on real and complex differentiability incl. applications. It therefore may not follow the usual division lines between courses and textbooks, but it is coherent in its own way and (almost, see below) self-contained. It originated in courses taught by the author at the Université Paris 6 for 3rd year licence students. The first part (chapters 1—8) starts from scratch and covers in a very well-structured way the basic notions of topology and functional analysis : real numbers, metric spaces, compact spaces, complete spaces, connected spaces, spaces of continuous functions, normed spaces, and Hilbert spaces. In these last two chapters, topics like the Theorems of Hahn-Banach and Banach-Steinhaus or the notion of adjoint operator are included. No knowledge of Lebesgue integration is assumed (and L^p-spaces therefore not discussed). This part summarizes neatly the sine qua non on these topics that every math student should master.

Part two (chapters 9—19) covers differentiable functions in one variable (with values in the reals or in a Banach space), functions differentiable on open subsets of Banach spaces, second derivatives, implicit functions, theorems for constant ranks, extremals, holomorphic functions, the residue theorem, convergence of sequences of holomorphic functions (Montel’s Theorem), maximum principle, conformal maps. The first chapters are probably written with applications in functional analysis in mind. The chapter on extremals can only touch the calculus of variations, since the book doesn’t cover differential equations. The chapter on holomorphic functions assumes some knowledge of 1-forms and contour integrals, though the principal results are quickly surveyed. Every chapter includes a list of class-room tested exercises, with some hints at the end of the book. Four appendices contain additional material on countable sets, the open mapping theorem, the Riemann sphere and the fixed point theorems of Brouwer and Schauder. Appendix E yields numerous examples of a French teaching specialty, namely, a list of « problèmes’’. These are longer exercises circling around a deeper topic and suitable for advanced several-hours-tests, vacation homework assignments or other opportunities where one needs to go beyond the usual exercises. In summary, the book is a nice addition to the existing literature, and will certainly be useful for students in French speaking countries. Last but not least, I would like to emphazise the very friendly price for a book of this size.

[Ilka Agricola (Berlin)]

Werner Kleinert (Berlin)**Hindry, Marc**. Arithmetics. Primality and Codes, Analytic Number Theory, Diophantine Equations, Elliptic Curves. (Arithmétique. Primalité et codes, théorie analytique des nombres, équations diophantiennes, courbes elliptiques.) (French)

[B] Paris : Calvage et Mounet. xvi, 328 p. EUR 40.00 (2008). ISBN 978-2-916352-04-6/hbk

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 Z/nZ, the finite fields 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 :

- 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.
- 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.
- Completions, p-adic numbers, Hensel’s lemma, the Hasse principle and a brief introduction to adeles and ideles.
- Transcendental numbers and Diophantine approximation in the light of the fundamental results by Thue, Siegel, Roth, and Baker.
- 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.
- 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 !

[Werner Kleinert (Berlin)]