## Reviews of André Weil's books

**1. Sur les espaces à structure uniforme et sur la topologie générale (1937), by André Weil.**

1.1. Review by J H C Whitehead.

The Mathematical Gazette23(255) (1939), 318-319.The object of this book is to abstract the notion of uniformity, as applied to continuity and convergence, from that of distance. Of course the topology of a metric space is not a metric geometry, like Euclidean or Riemannian geometry, since two metrics are regarded as equivalent if they lead to the same definition of closed sets. But the possibility of defining a given topological structure by means of a metric is a restriction. ... The book is clearly and attractively written and is an important contribution to axiomatic "point-set", as apart from combinatorial, topology.

**2. L'intégration dans les groupes topologiques et ses applications (1940), by André Weil.**

2.1. Review by: Shizuo Kakutani.

Mathematical Reviews, MR0005741(3, 198b).The author discusses those problems in the theory of topological groups which center around the notion of Haar measure. It is a clear exposition of the present state of the theory, and contains most of the important results which were recently obtained in this field. These results are completed in many points, and the methods of proof are greatly improved. On almost all pages the reader will find new ideas of the author. The book consists of seven chapters and two appendices. At the end of each section there are historical remarks, which, together with the complete references at the end of the book, give a clear and comprehensive view of the recent development of the theory.

**3. Foundations of algebraic geometry (1946), by André Weil.**

3.1. Review by: Daniel Pedoe.

The Mathematical Gazette31(297) (1947), 293-294.Algebraic geometry is that branch of mathematics which deals with the geometrical interpretation of algebraic equations. It has been universally recognised as an important and attractive branch of mathematics, but many mathematicians have been prevented from cultivating it by the feeling that its principles and methods are only fully understood by a small number of people, and that the novice wishing for initiation must undergo a long training in the avoidance of clear-cut (if laborious) algebraic methods, and in the development of his geometrical intuition. ... Andre Weil explains in his preface that his book arose "from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of characteristic p ≠ 0 (in which there is no transcendental method to guarantee the correctness of the results obtained by algebraic means), this being required for the solution of a long outstanding problem, the proof of the Riemann hypothesis in function-fields ". It is emphasised that " the main purpose of the book is to present a detailed and connected treatment of the properties of intersection-multiplicities, which is to include all that is necessary and sufficient to legitimise the use made of these multiplicities in classical algebraic geometry, especially of the Italian school ". ... it must be said that the book is beautifully written, and a remarkable piece of mathematics. The author has gone to great trouble to guide the reader, to explain what he is doing at every stage, and finally brings him, slightly dazed perhaps, to the frontier of present-day knowledge in certain branches of algebraic geometry. This is obviously a book which deserves and will repay careful study.

3.2. Review by: William Vallance Douglas Hodge.

Science. New Series107(2768) (1948), 75-76.The formulation of the theory of algebraic varieties on a strictly algebraic foundation, begun by van der Waerden and developed in a more advanced sphere by Zariski, is a development of profound significance for algebraic geometry and, indeed, for all mathematics, and the present volume, as the first book on the subject apart from van der Waerden's 'Einführung in die algebraische Geometrie', is an event of great importance. The developments referred to have three main objectives: first, to lay the foundations of the subject on a rigorous basis; secondly, to extend its scope; and thirdly, to provide the researcher with more powerful weapons than he has hitherto had at his disposal. ... Here the whole subject is conceived of as a development of the theory of fields, and the geometry appears only as an interpretation ... Because of the austere form in which the subject is presented and because the presentation is extremely condensed, theorem following theorem in seemingly endless procession, it is to be feared that many useful recruits to this kind of geometry will be frightened off rather than attracted. Indeed, the expert will not find it easy going and may easily miss much of value on a first reading. But, if he keeps returning to it, he will realise more and more how much of importance, not only for the purposes of the volume but for wider applications, is crowded into brief statements and proofs. As readers become more and more familiar with the work, it will come to be recognised for what it is - one of the real landmarks in the literature of algebraic geometry.

3.3. Review by: Otto F G Schilling.

Mathematical Reviews, MR0023093(9, 303c).Advances in the more arithmetic branches of modern algebra and their application to number theory naturally lead, as we may venture to say today, to problems which to the well-informed mathematician either appeared familiar as part of the heritage of classical algebraic geometry or seemed to be intrinsically adapted to a solution by more conceptual geometric methods. Furthermore, since major parts of the theory of algebraic functions of one variable had been fitted into the system of algebra it was sensible that similar interpretations and attempts at solutions were (and had to be) tried for higher dimensional problems. In order to understand and appreciate the ultimate significance of this book the reader may well keep in mind the preceding twofold motivation for the interest in algebraic geometry. Classical algebraic geometry made free use of a type and mode of reasoning with which the modern mathematician often feels uncomfortable, though the experience based on a rich and intricate source of examples made the founders of this discipline avoid serious mistakes in final results which lesser men might have been prone to make. The main purpose of this treatise is to formulate the broad principles of the intersection theory for algebraic varieties. We find those fundamental facts without which, for example, a good treatment of the theory of linear series would be difficult. The doctrine of this book is that an unassailable foundation (and thereby justification) of the basic concepts and results of algebraic geometry can be furnished by certain elementary methods of algebra.

**4. Variétés abéliennes et courbes algébriques (1948), by André Weil.**

4.1. Review by: Shiing-Shen Chern.

Bull. Amer. Math. Soc.56(1950), 202-204.This is the second of a series of papers with which the author promised to follow his book, Foundations of algebraic geometry, American Mathematical Society, 1946. The first paper, entitled: 'Sur les courbes algébriques et les variétés qui s'en déduisent', is concerned in particular with the theory of correspondences of an algebraic curve and contains the author's proof of the Riemann hypothesis over a finite field. In a sense this study is generalized to the case of general abelian varieties in the present work. ... The author has not only generalized the classical theory into a more profound new theory, with new results, but has presented the results in such a way that the development seems most natural. Among other things the book gives ample justification of the struggle one has to go through in reading the Foundations of the author.

4.2. Review by: Otto F G Schilling.

Mathematical Reviews, MR0029522(10, 621d).This book is a continuation of the author's test of the comprehensive methods developed in his "Foundations of Algebraic Geometry", which was begun in his "Sur les courbes algébriques et les variétés qui s'en déduisent" with the proof of the Riemann hypothesis for function fields of one variable. Now the theory of Abelian functions and of the Jacobian variety attached to a curve is developed over arbitrary coefficient fields. All the classical results (i.e., one works over the field of all complex numbers) are generalized and new theorems enriching the classical theory are added.

**5. Introduction à l'étude des Variétés Kählériennes (1958), by André Weil.**

5.1. Review by: Harry Levy.

Science. New Series129(3356) (1959), 1136-1137.Weil has done mathematics a great service, for his introduction to the subject should stimulate many mathematicians toward a more active interest in this new area of mathematics. His employment of the techniques of modern algebra and topology is effective and elegant. Of particular interest to the classical algebraic geometer is his treatment, in the final chapter, of theta functions and abelian varieties.

5.2. Review by: Michael H Atiyah.

The Mathematical Gazette44(347) (1960), 78.The theory of complex manifolds, and more especially of Kähler manifolds, is a very flourishing branch of current mathematics. The fundamental pioneering work was done by Hodge twenty years ago, and his book on Harmonic Integrals has until now been the only one on the subject. Professor Weil's book is essentially a modernised version of the Hodge Theory. It contains nothing basically new, but it is more up to date in its point of view. ... This is a very elegant book and should provide an excellent introduction to the subject.

**6. Adeles and algebraic groups (1961), by André Weil.**

6.1. Review by: King Fai Lai.

Mathematical Reviews, MR07341777(85c:01004).This is a review of the 1982 reprint.

This important work when it first "appeared" in 1961 introduced the adèles into the study of arithmetic problems in algebraic groups. The question of Tamagawa numbers of algebraic groups was first systematically formulated here. The book is divided into two parts. The first part - Chapters I and II - deals with the geometry and measure on the space of adelic points of an algebraic variety. These materials are not easily available elsewhere. In the second part - Chapters III and IV - the author studies the zeta function of division algebras and central simple algebras and then uses the Poisson summation formula to calculate the Tamagawa numbers of "most" classical groups

**7. Basic number theory (1967), by André Weil.**

7.1. Review by: George Whaples.

Mathematical Reviews, MR0234930(38 #3244).This is an exposition of algebraic number theory and class field theory, handling in a unified manner all fields, called by the author A-fields, which are finite algebraic over the rational field or over a field of rational functions of one variable with a strictly finite field of constants. The author states in the foreword that he has tried "to draw the conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number theory" and to show that from the point of view which he has adopted one could give a coherent account, logically and aesthetically satisfying, of the topics he was dealing with.

**8. Essais historiques sur la théorie des nombres (1975), by André Weil.**

8.1. Review by: Donald Maurer.

Mathematical Reviews, MR0389725(52 #10556).These lectures were transcribed, "with very little editing", from a tape recording. The discussion is leisurely, taking time for some interesting historical remarks about Fermat, Euler and others. The principal topics are (1) the connection between elliptic curves and number theory and (2) the zeta-function and its functional equation, starting with Fermat and Euler, respectively.

**9. Elliptic Functions According to Eisenstein and Kronecker (1976), by André Weil.**

9.1. From the Preface.'When kings are building', says the German poet, 'carters have work to do'. Kronecker quoted this in his letter to Cantor of September 1891, only to add, thinking of himself no doubt, that each mathematician has to be king and carter at the same time. But carters need roads. Not seldom, in the history of our science, has it happened that a king opened up a new road into the promised land and that his successors, intent upon their own paths, allowed it to be overrun by brambles and become unfit for transit. To help clean up such a road is the purpose of this little book, arising out of lectures given at the Institute for Advanced Study in the Fall of 1974 .... Where the road will lead remains, in large part, to be seen, but indications are not lacking that fertile country lies ahead.

9.2. Review by: Sarvadamam Chowla.

Mathematical Reviews, MR0562289(58 #27769a).In the reviewer's opinion this brilliant little book should be read alongside the author's monograph

Essais historiques sur la théorie des nombre, the best introduction to exciting fields of number theory. Many of these are intimately connected with the theory of the so-called "elliptic functions". These theories were developed by Legendre, Gauss, Jacobi, Eisenstein, Kronecker and many others in the last century. Today this field has grown into the theory of "modular forms". Its growth is made manifest by numerous conferences all over the world and is recorded, for example, in the "Lecture notes" series of Springer-Verlag.

**10. Number theory for beginners (1979), by André Weil.**

10.1. Review by: Terence Jackson.

The Mathematical Gazette64(429) (1980), 219-220.This book is based on notes for a ten-week course in number theory which the author taught at the University of Chicago in 1949. The course was an introductory one with a strongly algebraic slant, but as the author says of the notes, "Experience seems to show that they should be somewhat diluted for class use". ... The material in the book is very stimulating but I think that a reader unversed in abstract algebra would find it rather hard. ... I think most people who read this book will enrich and widen their knowledge, but it is not for absolute beginners.

10.2. Review by: J B Roberts.

Mathematical Reviews, MR0532370(80e:10004).This volume is a rather concise introduction to many of the more important topics of a first course in number theory. The details employ more use of the elements of group theory than is commonly found in such courses. Nevertheless, the book is self-contained and each chapter has a problem set. It would be particularly useful as a supplementary text for a first course in number theory or for self-study by someone with some mathematical maturity, but unfamiliar with number theory.

**11. Number theory. An approach through history: From Hammurapi to Legendre (1983), by André Weil.**

11.1. Review by: Eric Reyssat.

Revue d'histoire des sciences41(2), Algèbre, Analyse, Topologie. Questions d'histoire et d'interprétation (1988), 218-220.The history of number theory is so vast that to view it seems impossible without flying over it at a vast height or getting lost in the jungle. Weil proposes a third solution: in less than 400 pages, a guided sea voyage with overnight stops. How better to understand the birth and development of the law of quadratic reciprocity or the theory of elliptic integrals than by watching Fermat and Euler at work on a multitude of examples?

11.1. Review by: Ronald S Calinger.

Isis77(1) (1986), 153-154.In 'A Mathematicians's Apology' (1940), the English number theorist G H Hardy quoted his friend J E Littlewood as saying that the ancient Greek geometers "are not clever schoolboys or scholarship candidates, but Fellows of another college." In Number Theory, Andre Weil looks penetratingly at the accomplishments of leading "Fellows" who investigated the properties of integers prior to Karl Gauss. Weil is no detached observer. His book exudes his passion for number theory and his mastery of it. ... Two thirds of Weil's book deals with Fermat and Leonhard Euler. The detailed examination of Fermat's assertions and proofs rests on extant writings of Fermat and his correspondents and benefits from Weil's acute mathematical instinct. ... Weil excels in his analysis of Euler's achievements in number theory. ... Weil's book is aimed at a mathematically sophisticated audience; proofs are given in appendixes, and modern symbolism is used throughout. When it comes to the treatment of historical matters, the choice of modern symbolism has a drawback in that it obscures the difficulties encountered in stating and solving problems from earlier times. Nevertheless, his book adds greatly to our critical knowledge of the conceptual and experimental development of number theory to Gauss. It should long remain the standard work in this field.

11.2. Review by: Harold G Diamond.

Science. New Series226(4682) (1984), 1412.André Weil has prepared an incisive and well-written account of the development of number theory. His book is divided into four chapters and 11 appendices. The chapters are (i) a quick trip through the ancient world, with particular attention to the contributions of the Mesopotamians, Greeks and Indians; (ii) a visit with Fermat (1601-1665); (iii) an extended stay with Euler (1707-1783); and (iv) brief stops to see Lagrange (1736) and Legendre (1752-1833). ... Weil's book is not light reading in the vein of E T Bell's Men of Mathematics. It does. however, present such a wealth of material so well that it should have appeal to people with varying degrees of interest in number theory, and its appearance is to be warmly welcomed.

11.3. Review by: Paulo Ribenboim.

American Scientist73(5) (1985), 489.André Weil, one of the leading mathematicians of this century, has profoundly influenced the development of algebraic geometry and number theory, especially the area where these disciplines overlap. His lifelong familiarity with the history of number theory culminates in this book, which will be regarded as classical. Need less to say, it is thoroughly documented and the essential topics are covered. It is also written in a style which is precise yet flowing and free of pedantry. But in my opinion, the main strength of the book lies in the penetrating analysis of the thoughts and achievements in number theory of Fermat, Euler, Lagrange, and Legendre. Weil succeeds very well in situating them in the historical and scientific context. The reader is made aware of the state of number theory in their times, of which methods were available to them, and of how the problems they faced, and sometimes solved, fit into the general development of number theory. ... In short, this is an extraordinary book - lively, essential, and authoritative.

11.4. Review by: Ezra Brown.

Mathematical Reviews, MR0734177(85c:01004).This book is a study of various number-theoretic texts ranging from the Babylonian tablet known as Plimpton 322 to A M Legendre's Essay on the theory of numbers, and covering a time span from ca. 1900-1600 B.C. to A.D. 1798 - a span of roughly three and a half millenia. As the author says, this is a historical treatment of that oldest and purest field of mathematics, the theory of numbers; his presentation is meticulous and scholarly. ... In the introduction, the author says that "no specific knowledge is expected of the reader, and it is the author's fond hope that some readers at least will find it possible to get their initiation into number theory by following the itinerary retraced in this book". This is an extremely optimistic viewpoint, for the book is not always easy to read, and in some places the going is rough. In all fairness, the author does recommend, for background, his own "Number theory for beginners" or Chapter 1 from Serre's "A course in arithmetic". This is a wise suggestion: even better, the interested reader should have a term of number theory under his or her belt before tackling Professor Weil's intriguing contribution to the history of number theory. ... The volume ... is a discursive, expository, leisurely peek over the shoulders of several great authors in number theory, a subject "conspicuous for the quality rather than for the number of its devotees; at the same time it is perhaps unique in the enthusiasm it has inspired", as Professor Weil says in his preface.

**12. Correspondance entre Henri Cartan et André Weil (1928-1991), by Andre Weil and Henri Cartan.**

12.1. Review by: Solomon Marcus.

Mathematical Reviews, MR2866913.H Cartan and A Weil were most often geographically far away from each other; Cartan was almost permanently in France, while Weil was involved in the itinerary Paris-Rome-Göttingen-Berlin-Stockholm-India (1930-1932), Marseille-Strasbourg-Finland-Sweden-France (partly in prison in the three countries), then in the USA, 1940-1945, in São Paulo (Brazil), 1945-1947, after which he moved again to the USA: Chicago, 1947-1958, and Princeton, 1958-1998. Having the same scientific start, at the École Normale Supérieure (Paris), they became friends, sharing similar research interests. Their need of interaction, in a period before the emergence of the internet and of e-mail, had been satisfied by using the traditional post. During more than 60 years, they wrote letters in which they described their questions, their attempts, their failures, their doubts, and their findings, but they were also ready to pay attention to the partner's problems and findings. The reader has the rare opportunity to look into the working laboratory of two of the most important scholars of the 20th century. A large part of their letters is devoted to commenting on the ideas and results belonging to other mathematicians that are related to Cartan and Weil's common interests.

JOC/EFR November 2014

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