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André Weil: Oeuvres Scientifiques - Collected Papers I (1926-1951)

André Weil
Publication Date: 
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Springer Collected Works in Mathematics
[Reviewed by
Michael Berg
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When I was a student many decades ago, in the days when, after an unquestionably spectacular trajectory, André Weil’s active career was drawing to a close, I was subjected to a number of tales about him that painted him as something of a terror. I guess the first story I heard concerned an episode, presumably at Chicago (where he was a mainstay just prior to his long-time residence at the Institute of Advanced Study), where his interlocutor asked him who, in his opinion, was the best mathematician in the world. Without hesitating Weil said, “Carl Ludwig Siegel.” The follow-up question, of course, asked for the next best one; apparently Weil just smiled and polished his fingernails against the lapel of his coat. Formidable. And then there is what is now certainly my favorite story, concerning Weil’s evident leadership role in a very sporty opposition to a candidate for a chair in economics (as I recall) at IAS who had been proposed by the then-director of the Institute, J Robert Oppenheimer. Evidently the man had made his mark with an analysis of certain labor conditions at a shoe-factory, and Weil et al seized on this as a sufficient condition for refusing to endorse the appointment as beneath the dignity of the Institute. An exasperated Oppenheimer commented, in phase with G. H. Hardy’s famous estimate (in A Mathematician’s Apology), that mathematicians, as a species, work at their incomparably demanding craft for on the order of four hours daily, after which they cannot go on: it’s simply too exhausting (Grothendieck being the great Gegenbeispiel, of course), and this makes them very angry. Oppenheimer argued that it was this element of frustration, and its resulting ill temper, that translated into his Institute’s mathematicians’ revolt. Subsequently this went on to take on far broader dimensions: evidently it is a matter of record that Oppenheimer was frequently at odds with the pure mathematicians.

Well, this brings up a number of points. Weil was nothing if not a rebellious non-conformist who, even as a mandarin, would not give up his Normalien ways. Even as he was, from very early on in his stellar career, a superb and masterful mathematician, he was inevitably opposed to the status quo, from his choice of a thesis topic alien to his professors at l’École Normale Supérieure to his long-time role in Bourbaki, of which he was in fact a founder. He was an internationalist early on, travelling as a very young man to Germany, including to Göttingen (and becoming a master of Riemannian geometry “at the fount,” as it were), teaching in India as well as São Paulo, Brazil, before coming to and settling in the United States, all the while remaining heavily involved in French mathematics. As a member of Bourbaki he occasionally listed his academic affiliation as the University of Nancago (= a linear combination of Nancy and Chicago) but, throughout it all, Weil indeed grew into one of the undisputed premier mathematicians of the twentieth century.

In due course, then, when he took up residence in the US, he first became one of the main figures at the University of Chicago, and thereafter, as already indicated, a Professor at the Institute for Advanced Study. There is no question that he was one of the major forces behind the magnificent evolution of algebraic geometry, as well as arguably the most dominant player of all in the theory of numbers in its modern sense. Specifically, Weil introduced a collection of themes into these fields that are today a major part of every one’s arsenal: as far as number theory alone is concerned, consider, e.g., the ubiquity of adèlic methods (I was told once that Weil named his construction the ring of adèles in homage to a girl-friend’s name) and the very subject of automorphic representations. On this latter topic it is certainly the case that there are many claimants to the title of progenitor, but there is no question that Weil’s writings, including the terse but critical early text, l’Intégration dans les Groupes Topologiques et ses Applications, were definitive, and his articles extending the classical works by Hecke and by Siegel are properly counted as revolutionary: they largely inaugurated the approach through representation theory.

In the latter connection, the article, Sur Certains Groupes d’Opérateurs Unitaires, published in 1964 in Acta Math. (and found on p. 1 of Vol. III of the set of books under review), is among the single most important works in the area of the application of (abstract) Fourier analytic methods to questions in number theory. In this beautiful but very dense paper Weil presented the analytic theory of quadratic forms, as developed not long before by C. L. Siegel, in terms of the unitary representation theory of the symplectic group, introducing in the process the famous projective representation that is now usually called the Weil representation and recasting the proof of quadratic reciprocity for a number field (first given by Erich Hecke about forty years earlier using classical Fourier analysis and theta series) in the indicated representation theoretic form. V. S. Varadarajan, who was my undergraduate mentor at UCLA and with whom I did a sabbatical in the 1990s, characterized this article to me as “Weil’s great Acta paper.”

A propos, back in the late 1970s Varadarajan gave me a reading course using Weil’s classic Elliptic Functions According to Eisenstein and Kronecker and in the early 1980s I did my PhD thesis with Audrey Terras at UCSD on Weil’s so-called inverse Hecke correspondence. Accordingly I was indeed exceedingly fortunate to have been exposed to both Weil’s mathematics and his style early on. I think it is fair to say that he is not easy to read, but with some digging you unearth jewel after jewel.

If I may continue on a personal note (aimed also at giving the lie to the prevailing impression that for all his brilliance Weil was an unkind man), it was in connection with Sur Certains Groupes d’Opérateurs Unitaires that I first corresponded with Weil. Having just worked though Hecke’s Vorlesungen über die Theorie der algebraischen Zahlen and learnt about his challenge to extend his proof of quadratic reciprocity to higher degrees, I contacted a number of prominent scholars in number theory, Weil among them, to get their advice on what to do. In about two weeks I received a note from Weil, typed on his famous typewriter, encouraging me and pointing me toward Sur Certains Groupes d’Opérateurs Unitaires. I subsequently spent a wonderful time working carefully through much of this text and can safely say, with gratitude, that many of my own publications are concerned with what I learned from there. Over the next few years, as a rookie in the academic game, I had two more letter exchanges with Weil, now already in his late eighties. I am forever grateful to him.

So, the three books under review here, i.e. Weil’s Collected Papers, are certainly meaningful to me on an obvious personal count; however, well beyond this, these books contain a number of masterpieces and present example after example not only of Weil’s unique artistry, but they reveal a great deal about the man, including his critical sense about everything from the way mathematics should be presented to the way it should be learned. I have long tried to take his maxim to the effect that when learning something mathematical one should make for the center as soon as possible and not spend too much time crawling around the edges (see p. 121 of Vol II for the exact quote). I’m afraid that I generally fail at this: I tend to get neurotically fascinated by minutiae situated at the edges: dotting all the i’s and crossing all the t’s can become a vice. But Weil is a wonderful guide, and he certainly provides rules to work by in these volumes.

But it is the mathematics proper, of course, which is the real game, and these Oeuvres are a treasure. The books are delineated by time demarcations: Vol. I covers 1926–1951, Vol. II covers 1951–1964, and Vol. III covers 1964–1978. Weil passed into eternity in 1998, at 92, so he was formally active as a mathematician for a very long time. In very rough terms, which to an extent do him a disservice, the first volume can be said to concern primarily differential geometry (including Riemannian geometry), algebraic geometry in a somewhat earlier form (remember that Serre and Grothendieck should count as members of the next generation to Weil’s, although we do find Weil’s 1941 paper on the Riemann hypothesis for function fields here on p. 277), and class field theory. (Yes, there’s more, but I’ll take the liberty to refer to the table of contents.)

The second volume shows a movement in the direction of more modern algebraic geometry (remember that Weil was one of its proto-architects) as well as fascinating developments in algebraic number theory. Indeed the adèles begin to figure more and more prominently and a true modern spirit is becoming manifest. Finally, in the third volume (my own favorite) we encounter not only a number of major works that evince a synthesis of earlier themes, e.g., Weil’s application of a number of marvelous analytic and representation theoretic tools with which to approach large swaths of mathematics from a unified perspective, disclosing deep connections and presenting both new problems and new areas of development, but also a clear historical orchestration that adds structure to it all.

I want to draw explicit attention to two articles of particular import, namely, the aforementioned Sur Certains Groupes d’Opérateurs Unitaires and his 1967 masterpiece, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, on p. 165 ff. of Vol. III, because they encapsulate one of the main themes of Weil’s life-long scholarship. Throughout, from his early focus on the work of Riemann to his later works building magnificently on the ideas and constructs of, for example, Hecke and Siegel (the 1967 paper just cited in fact concerns the earlier “inverse Hecke correspondence” between automorphic functions and Dirichlet series: the Mellin transform formalism rules the roost — see also Weil’s 1968 paper “Zeta-functions and Mellin transforms,” on p. 179 ff. of loc. cit.), André Weil was influenced by a deep sense and understanding of the historical forces at work in the development of mathematics and how these conspire to point toward the future, and this is, to be sure, was a major factor behind the depth of his many contributions.

A final caveat: in order truly to appreciate the contents of this set of books, the reader should be able to read all three languages Weil was fluent in and wrote so beautifully: his native French, and then more or less chronologically relative to his life, German and English. Happily his prose in all three languages is beautifully limpid, even if his mathematics is dense: but the density of the mathematics should slow the reader down somewhat, better to be able to appreciate the prose.

In closing, let me recommend two poignant biographical works. First, Weil’s autobiographical The Apprenticeship of a Mathematician (1991) and the book At Home with André and Simone Weil, the 2001 work written by Weil’s daughter, Sylvie. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Exposés au séminaire Bourbaki

Sur les surfaces à courbure negative
Sur les espaces fonctionnels
Sul calcolo funzionale lineare
L’arithmétique sur une coube algébrique
L’arithmétique sur les courbes algébriques
Sur un théorème de Mordell
On systems of curves on a ring-shaped surface
Sur les series de polynomes de deux variables complexes
Une théorème d’arithmétique sur les courbes algébriques
Über das Verhalten der Integrale erster Gattung bei Automorphismen des Funktionenkörpers
Une propriété charactéristique des groupes de substitutions linéaires finis
Une propriété charactéristique des groupes finis de substitutions
Über Matrizenringe auf Riemannischen Flächen und den Riemann-Rochschen Satz
Arithmétique et Géométrie sur les variétés algébriques
Sur les fonctions presque périodiques de von Neumann
L’intégrale de Cauchy et les fonctions de plusieurs variables
Démonstration topologique d’un théorème fondamental de Cartan
Les familles de courbes sur le tore
Arithmetic on algebraic varieties
Mathematics in Indian Universities
Mathematics in India
La mesure invariante dans les espaces de groups et les espaces homogènes
La théorie des enveloppes en Mathématiques Spéciales
Les recouvrements des espaces topologiques: espaces complets, espaces bicompacts
Sur les groupes topologiques et les groupes mesurés
Sur les fonctions elliptiques p-adiques
Remarques sur des resultats récents de C. Chevalley
Sur les espaces à structure uniforme et sur la topologie générale
Généralization des fonctions abéliennes
Zur algebraischen Theorie der algebraischen Funktionen
“Science Française”
Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques
Sur les groupes à pn éléments
Une lettre et un extrait de lettre à Simone Weil
Sur les fonctions algébriques à corps de constants fini
Calcul des probabilités, méthode axiomatique, integration
L’Intégrations dans les groupes topologiques et ses applications (Dédicace et Introduction)
On the Riemann hypothesis in function-fields
Lettre à Artin
The Gauss-Bonnet theorem for Riemannian polyhedra
Differentiation in algebraic number-fields
A correction to my book on topological groups
Foundations of algebraic geometry (Introduction)
Sur quelques résultats de Siegel
L’avenir des mathématiques
Sur la theories des formes différentielles attachées à une variété analytique complexe
Sur les courbes algébriques et les variétés que s’en deduisent (Introduction)
Variétés abéliennes et courbes algébriques (Introduction)
On some exponential sums
Sur l’étude algébrique de certains types des lois de mariage
Numbers of solutions of equations in finite fields
Fibre-spaces in algebraic geometry
Théorèmes fondamentaux de la théorie des fonctions thêta
Géométrie différentielle des espaces fibrés
Variétés abéliennes
Number-theory and algebraic geometry
Arithmetic on algebraic varieties
Sur la théorie du corps de classes