## Reviews of Jean-Pierre Serre's books

**1. Groupes algébriques et corps de classes**(1959)

**, by Jean-Pierre Serre.**

**1.1. Review by: Masayoshi Nagata.**

*Mathematical Reviews*, MR0103191

**(21 #1973)**.

This is mainly an exposition of the theory of generalized Jacobian varieties by Maxwell Rosenlicht and the class field theory over function fields by Lang. Very little knowledge (mainly the sheaf-theoretic definition of algebraic varieties by Serre) is pre-assumed, so that this will be an easy text book on these subjects.

**2. Corps locaux (1962), by Jean-Pierre Serre.**

**2.1. Review by: Tadashi Nakayama.**

*Mathematical Reviews*, MR0150130

**(27 #133)**.

The book grew out of a course of lectures at the Collège de France (1958-1959), and expounds local class field theory and related subjects. Its fifteen chapters are grouped into four parts. The first part starts by characterizing discrete-valuation rings (e.g., as Noetherian local rings with principal and non-nilpotent maximal ideal). Dedekind rings and their extensions are considered; norm and injection of (fractional) ideals are discussed ... The second part is about ramification ... The third part, the longest among the four, is on cohomology ... The fourth part starts with Witt's theorem on the Brauer group ... Then, local class field theory is developed ... Homological methods are eminent throughout the book. Geometric view-points are marked.

**3. Cohomologie galoisienne (1964), by Jean-Pierre Serre.**

**3.1. Review by: Marvin Jay Greenberg.**

*Mathematical Reviews*, MR0180551

**(31 #4785)**.

This book surveys an elegant new subject which has developed out of the cohomological treatment of class field theory by Emil Artin and John Tate. The bulk of the early contributions were by Tate, and we are greatly indebted to the author for publishing them in his very lucid style. Many others have made impressive discoveries in the field since ...

**3.2. From the Foreword.**

This relates to the English translation *Galois cohomology* (1997).

This volume is an English translation of *Cohomologie galoisienne*. The original edition was based on the notes, written with the help of Michel Raynaud, of a course I gave at the Collège de France in 1962-1963. In the present edition there are numerous additions and one suppression: Jean-Louis Verdier's text on the duality of profinite groups. The most important addition is the photographic reproduction of a paper of Robert Steinberg's "Regular elements of semisimple algebraic groups" (1965). Other additions include: - A proof of the Golod-Shafarevich inequality - The "resume de cours" of my 1991-1992 lectures at the College de France on Galois cohomology of *k*(*T*) - The "resume de cours" of my 1990-1991 lectures at the College de France on Galois cohomology of semisimple groups, and its relation with abelian cohomology, especially in dimension 3. The bibliography has been extended, open questions have been updated (as far as possible) and several exercises have been added.

**4. Algèbre Locale. Multiplicités (1965), by Jean-Pierre Serre.**

**4.1. Review by: Masayoshi Nagata.**

*Mathematical Reviews*, MR0201468

**(34 #1352)**.

This article exposes a theory of local rings, mostly the equal characteristic case. Some general observations on noetherian modules and their primary decomposition are made in Chapter I. Then, in Chapter II, filtration, gradation and Hilbert polynomials are considered. The main topics in Chapter III are (1) basic theorems on integral extensions and Krull dimension (including the going up theorem and the going down theorem of Cohen-Seidenberg, the altitude theorem of Krull and the notion of a system of parameters), (2) characterization of normality of a noetherian integral domain (due to Krull), and (3) normalization theorem for a finitely generated ring over a field and its applications. The main topics in Chapter IV are (1) applications of the Koszul complex to the theory of local rings (originated by Auslander, Buchsbaum, and the author), (2) homological dimension and homological co-dimension, (3) Cohen-Macaulay modules, (4) homological characterization of a regular local ring (by finiteness of homological dimension), and (5) structure theorem of a complete local ring (only for the equal characteristic case).

**4.2. Publisher's description.**

This relates to the English translation *Local algebra* (2000).

This is an English translation of the now classic "Algèbre Locale - Multiplicités" originally published by Springer. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities. Many modifications to the original French text have been made for this English edition, making the text easier to read, without changing its intended informal character.

**4.3. From the Preface.**

This relates to the English translation *Local algebra* (2000).

The English translation, done with great care by CheeWhye Chin, differs from the original in the following aspects: (1) The terminology has been brought up to date (e.g. 'cohomological dimension' has been replaced by the now customary 'depth'). (2) I have rewritten a few proofs and clarified (or so I hope) a few more. A section on graded algebras has been added (Appendix III to Chapter IV). (3) New references have been given, especially to other books on commutative algebra: Bourbaki (whose Chapter X has now appeared, after a 40-year wait), Eisenbud, Matsumura, Roberts, ....

**5. Lie Algebras and Lie Groups (1965), by Jean-Pierre Serre.**

**5.1. From the Introduction.**

Part I. Lie algebras. The main general theorems on Lie algebras are covered, roughly the content of N Bourbaki's *Éléments de mathématique* Chapter I [*Groupes et algèbres de Lie, Chapitre* 1: *Algèbres de Lie*]. I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-*p*-groups.

Part II. Lie groups. This part is meant as an introduction to formal groups, analytic groups, and the correspondence between them and Lie algebras (Lie theory). Analytic groups are defined over any complete field (real, complex, or ultrametric); Lie theory applies equally well to both these cases, provided the characteristic is zero.

**5.2. Review by: C Terence C Wall.**

*The Mathematical Gazette* **51** (375) (1967), 76-77.

This book gives a comprehensive and modern account of the main results concerning Lie algebras, and their relation to formal groups, local Lie groups (here called group chunks), and Lie groups. It is unusual among books giving accounts of this relation in that the author constantly draws on results from different branches of mathematics, and then emphasises the applications of his results to several different fields. Consequently, the book is not suitable for a reader with a narrow mathematical background. The pace is brisk throughout; proofs are shortened by efficient notations, abstraction is neither sought for its own sake nor shirked when it would simplify the argument, and references are given to Bourbaki when a substantial result is quoted. Trivial steps in proofs are left to the reader; the author's judgment on this is good, but occasional difficulty will be experienced. Several exercises are proposed; and seem to be well-chosen.

**6. Algèbres de Lie semi-simples complexes (1966), by Jean-Pierre Serre.**

**6.1. Review by: Shingo Murakami.**

*Mathematical Reviews*, MR0215886

**(35 #6721)**.

This book consists of notes of the author's lectures at Algiers in 1965 and provides a concise presentation of the theory of complex semisimple Lie algebras. Actually, the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book.

**7. Abelian l-adic representations and elliptic curves (1968), by Jean-Pierre Serre.**

**7.1. Review by: Kenneth Alan Ribet.**

*Bull. Amer. Math. Soc. (N.S.)*

**22**(1) (1990), 214-218.

Addison-Wesley has just reissued Serre's 1968 treatise on *l*-adic representations in their Advanced Book Classics series. This circumstance presents a welcome excuse for writing about the subject, and for placing Serre's book in a historical perspective. ... Despite recent developments, the 1968 book of Serre is hardly outmoded. For one thing, as the cover of the new edition reminds us, it's the only book on the subject. More importantly, it can be viewed as a toolbox which contains clear and concise explanations of fundamental facts about a series of related topics: abstract *l*-adic representations, Hodge-Tate decompositions, elliptic curves, *L*-functions, etc.

**7.2. Review by: Takashi Ono.**

*Mathematical Reviews*, MR0263823 **(41 #8422)**.

There are four chapters. The first two chapters reflect the work of Yutaka Taniyama ... Chapter I contains definitions and examples of *l*-adic representations ... Chapter II gives the construction of some abelian l-adic representations. As mentioned above, this is essentially due to Yutaka Taniyama. ... In Chapter III, the author conjectures that every rational semi-simple abelian *l*-adic representation ... should come from an algebraic representation ... in the manner described in Chapter II. Chapter IV is concerned with the representation for an elliptic curve ...

**8. Représentations linéaires des groupes finis (1968), by Jean-Pierre Serre.**

**8.1. Review by: Tsit-Yuen Lam.**

*Mathematical Reviews*, MR0232867

**(38 #1190)**.

As stated in the introduction, the three parts comprising this book on representation theory differ considerably in level as well as in objectives. The first part entitled "Representations and characters'' was to be the appendix of a book on quantum chemistry, authored by Gaston Berthier and Josiane Serre. Here, representations of finite groups are studied exclusively over the complex numbers, thus avoiding considerations of bad characteristic and ground field extensions. The classical theory of Schur and Frobenius is then pleasantly displayed. ... The more penetrating features of representation theory are then taken up in Part II.

**8.2. Publisher's description.**

Description of the English translation *Linear representations of finite groups* (1977).

This book consists of three parts, rather different in level and purpose. The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. The second part is a course given in 1966 to second year students of l'École Normale. It completes in a certain sense the first part. The third part is an introduction to Brauer Theory.

**8.3. Review by: William Howard Gustafson.**

*Bull. Amer. Math. Soc.* **84** (5) (1978), 939-943.

This review is of the English translation *Linear representations of finite groups* (1977).

As one so often discovers in tracing the history of a beautiful mathematical idea, we have Gauss to thank for pointing the way towards the representation theory of finite groups. In his discussion of class groups of even binary quadratic forms of given discriminant over the integers, he attempted to attach to each class what he considered to be its "character", in order to distinguish among genera. His description was rather cumbersome, so it fell to Dedekind, inspired by Dirichlet's notation for Gauss' characters, to give the more familiar interpretation of characters as numerical functions. The general notion of irreducible characters of abelian groups was set down in 1879, in the third edition of Dirichlet and Dedekind's *Vorlesungen über Zahlentheorie, *and the inclusion of the idea in Weber's famous algebra text made it familiar to many mathematicians of the late nineteenth century. ... Serre's book gives a fine introduction to representations for various audiences. It is divided in three parts. The first was originally an appendix to a book on quantum chemistry by Gaston Berthier and Josiane Serre. It gives an exposition of the basics of complex characters and representations, in a style suitable for non-specialists. There are also a few remarks on the extension of the theory to compact groups. The second part is for a more sophisticated reader. It gives more detailed information on complex characters, and then proceeds to deeper topics. ... The third part is an exposition of Brauer's modular theory. ... Despite the brevity of the book and its omission of many topics, the specialist can profit greatly from reading it. As always with Serre, the exposition is clear and elegant, and the exercises contain a great deal of valuable information that is otherwise hard to find. Also, the discussion of rationality questions is by far the best available.

**9. Cours d'arithmétique (1970), by Jean-Pierre Serre.**

**9.1. Review by:**

**Reuben Louis Goodstein.**

*The Mathematical Gazette*

**55**(393) (1971), 342.

This book in the SUP series is based upon two courses of lectures given by the author in 1962 and 1964 at the Ecole Normale Supérieure. The book is in two parts. The first part is purely algebraic and is concerned with quadratic forms over the field of rational numbers; amongst the topics discussed in detail are the law of quadratic reciprocity, *p*-adic fields and quadratic forms with discriminant ± 1. The second part uses function theoretic methods, and contains in particular on account of Dirichlet's proof of his famous theorem that every arithmetical progression contains an infinity of primes. The volume ends with a study of modular functions, and in particular of theta functions.

**9.2. Review by: Burton Wadsworth Jones.**

*Mathematical Reviews*, MR0255476 **(41 #138)**.

Part I of this monograph, consisting of the first five chapters, is purely algebraic and has as its object the classification of quadratic forms over the rational field. ... Part II is analytic in nature, Chapter VI giving the proof of Dirichlet's theorem on primes in an arithmetic progression and Chapter VII dealing with modular forms including Hecke's operators and a brief treatment of theta functions. These two parts correspond to courses given in 1962 and 1964 to second year students in l'École Normale Supérieure. While not a text in the American sense since is has no exercises, it is very readable in spite of moving quite quickly into rather deep mathematics. ... This is a notable contribution to the literature on quadratic forms.

**9.3. From the Preface.**

This extract is from the English translation *A Course in Arithmetic* (1973).

This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, *p*-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± 1. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomorphic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the École Normale Supérieure.

**9.4. Review by: Herve Jacquet.**

*American Scientist* **62** (1) (1974), 120.

This review is of the English translation *A Course in Arithmetic* (1973).

Although divided into two parts (algebraic methods and analytic methods), this remarkable book deals with three main topics: quadratic forms, Dirichlet's density theorem, and modular forms. ... The book is carefully written - in particular, very much self-contained. As was the intention of the author, it is easily accessible to graduate or even undergraduate students, yet even the advanced mathematician will enjoy reading it. The last chapter, more difficult for a beginner, is an introduction to contemporary problems.

**10. Arbres, amalgames,**

*SL*_{2}(1977), by Jean-Pierre Serre.**10.1. Review by: Ernest Borisovich Vinberg.**

*Mathematical Reviews*, MR0476875

**(57 #16426)**.

This book contains a part of a course given at the Collège de France in 1968-69. In the first chapter the author shows how we can reconstruct a group *G* operating on a tree *X* from *X*/*G* and stabilizers of vertices and edges. In particular, if these stabilizers are trivial, *G* is free so obtaining a simple proof of the theorem of Otto Schreier stating that any subgroup of a free group is free.

**10.2. From the English translation** *Trees* (1980).

The present book is an English translation of "Arbres, amalgames, *SL*_{2}", published in 1977 by Jean-Pierre Serre, and written with the collaboration of Hyman Bass. The first chapter describes the "arboreal dictionary" between graphs of groups and group actions on trees. The second chapter gives applications to the Bruhat-Tits tree of *SL*_{2} over a local field.

**10.3. Review by: Roger C Alperin.**

*Bull. Amer. Math. Soc. (N.S.)* **8** (2) (1983), 401-405.

Serre's book on trees is the exposition of the first part of his 1968-69 course on discrete groups. It begins *ab initio *with a leisurely study of the two main constructions of combinatorial group theory: free products with amalgamation and the HNN extension. These arise in topology, for example, in the description of the fundamental group of a 3-manifold which splits along a connected incompressible surface. Furthermore, in each of these two group-theoretic constructions there is a naturally defined tree, *T*, on which the group acts so that the quotient by the group action is an edge or a loop, respectively. In the topological situation this is just imitating the action of the fundamental group on the universal covering space. ... Although, in translation, *Trees *has lost its magnificent frontispiece and virtually tripled in price, it's still a great buy.

**11. Lectures on the Mordell-Weil theorem (1989), by Jean-Pierre Serre.**

**11.1. Publisher's description.**

The book is based on a course given by Jean-Pierre Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem.

**11.2. Abstract by Jean-Pierre Serre.**

The aim of this course is the study of rational and integral points on algebraic varieties, especially on curves or abelian varieties. Before the end of the last century only special cases had been considered. The first general results are found, around 1890, in the work of Hurwitz and Hilbert where they introduced the, nowadays natural, viewpoint of algebraic geometry ...

**11.3. Review by: Joseph Hillel Silverman.**

*Mathematical Reviews*, MR1002324 **(90e:11086)**.

This book, which covers far more material than is indicated by its modest title, is based on lectures given by the author at the Collège de France in 1980 and 1981. Michel Waldschmidt's hand-written notes of these lectures have enjoyed a wide circulation, so this edited and translated version by Martin Brown is very welcome. The subject matter of these lectures is Diophantine geometry, especially the theory of height functions and finiteness theorems for integral and rational points on curves and abelian varieties.

**12. Topics in Galois theory (1992), by Jean-Pierre Serre.**

**12.1. Publishers description.**

This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Arnold Scholz and Hans Reichardt construction for p-groups, as well as Hilbert's irreducibility theorem and the large sieve inequality, are presented. The second half is devoted to rationality and rigidity criteria and their application in realising certain groups as Galois groups of regular extensions of Q(T). While proofs are not carried out in full detail, the book contains a number of examples, exercises, and open problems.

**12.2. Review by: Michael Fried.**

*Bull. Amer. Math. Soc. (N.S.)* **30** (1) (1994), 124-135.

Serre's book is a set of topics. It contains historical origins and applications of the inverse Galois problem. Its audience is the mathematician who knows the ubiquitous appearance of Galois groups in diverse problems of number theory. Such a mathematician has heard there has been recent progress on the inverse Galois problem. Serre has written a map through the part of this progress that keeps classical landmarks in sight.

**12.3. Review by: B Heinrich Matzat.**

*Mathematical Reviews*, MR1162313 **(94d:12006)**.

This small book contains a nice introduction to some classical highlights and recent work on the inverse Galois theory problem. The topics and main theorems are carefully chosen and composed in a masterly manner. In part the proofs are only sketched, but in any case they give a good basic understanding for the main problems and results. ... Altogether, this book can be highly recommended to anyone interested in this very classical problem, in the methods so far developed to solve it and in the partial solutions obtained up to now.

**13. Lectures on**

*N*_{X}(*p*) (2011), by Jean-Pierre Serre.**13.1. Publisher's description.**

Lectures on *N*_{X}(*p*) deals with the question on how *N*_{X}(*p*), the number of solutions of mod *p* congruences, varies with *p* when the family (*X*) of polynomial equations is fixed. While such a general question cannot have a complete answer, it offers a good occasion for reviewing various techniques in *l*-adic cohomology and group representations, presented in a context that is appealing to specialists in number theory and algebraic geometry. Along with covering open problems, the text examines the size and congruence properties of *N*_{X}(*p*) and describes the ways in which it is computed, by closed formulae and/or using efficient computers. The first four chapters cover the preliminaries and contain almost no proofs. After an overview of the main theorems on *N*_{X}(*p*), the book offers simple, illustrative examples and discusses the Chebotarev density theorem, which is essential in studying frobenian functions and frobenian sets. It also reviews l-adic cohomology. The author goes on to present results on group representations that are often difficult to find in the literature, such as the technique of computing Haar measures in a compact *l*-adic group by performing a similar computation in a real compact Lie group. These results are then used to discuss the possible relations between two different families of equations *X* and *Y*. The author also describes the Archimedean properties of *N*_*X*(*p*), a topic on which much less is known than in the *l*-adic case. Following a chapter on the Sato-Tate conjecture and its concrete aspects, the book concludes with an account of the prime number theorem and the Chebotarev density theorem in higher dimensions.

**13.2. Review by: Gabor Wiese.**

*Mathematical Reviews*, MR2920749.

This book is an expanded version of lectures delivered by Abel prize winner Jean-Pierre Serre. It is about a fundamental question in mathematics: the number of solutions of polynomial equations. ... This can be seen both as a Diophantine and as a geometric problem. Accordingly, methods from many different areas, most notably algebraic geometry, algebraic and analytic number theory, and representation theory, enter the subject. ... The book is written by a master in the area. It puts the objects it treats into their natural conceptual framework. The book requires some knowledge of algebraic geometry, (algebraic and analytic) number theory, and representation theory. It can be read with great profit by graduate students and researchers. The book, having arisen from lecture notes, includes many exercises. The wealth of references to original papers from Galois to the most recent research results is a valuable resource for the working researcher. There are almost no proofs in Chapters 1 to 4; however, most proofs in the later chapters of the book are given or at least sketched. The book is highly recommended to anyone interested in the fundamental questions it treats. Those enjoying the mathematics created by Serre will also find pleasure and inspiration in this book.

JOC/EFR November 2014

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