VI.93 André Weil

b. Paris, 1906; d. Princeton, New Jersey, 1998

Algebraic geometry; number theory

André Weil was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the BOURBAKI [VI.96] group, of which he was one of the principal founders.

Weil, as well as his sister, the philosopher, political activist, and religious thinker Simone Weil, received an excellent education. Both were brilliant students, very widely read, with a keen interest in languages (including Sanskrit). André Weil soon specialized in mathematics, his sister in philosophy. He graduated (and was first in his year in the agrégation for mathematics) from the École Normale Supérieure (ENS) when he was not even nineteen years old, and traveled in Italy and Germany. He obtained his doctorate in Paris at the age of twenty-two, and then went to Aligarh, India, as a professor for two years. After a brief spell in Marseilles, he was Maître de Conférences at Strasbourg University (along with Henri Cartan) from 1933 to 1939. The idea of the Bourbaki project arose there from discussions about teaching with Cartan, and grew in Paris in meetings that included other friends from the ENS.

His research achievements began with his 1928 Paris thesis. In it, he generalized MORDELL’S THEOREM [V.29] of 1922, that the group of rational points on an ELLIPTIC CURVE [III.21] is a finitely generated Abelian group, to the group of K-rational points (where K is a NUMBER FIELD [III.63]) of a Jacobian variety. During the following twelve years, Weil branched out in various directions, all related to important research topics of the 1930s: the approximation of holomorphic functions of several variables by polynomials; the conjugation of maximal tori in compact LIE GROUPS [III.48 §1]; the theory of integration on compact and Abelian topological groups; and the definition of uniform TOPOLOGICAL SPACES [III.90]. But problems of arithmetic origin stood out among his interests: further thoughts on his thesis and on Siegel’s finiteness theorem for integral points; a bold “vector bundle” version of the RIEMANNROCH THEOREM [V.31] on a Riemann surface (in parallel with similar work by E. Witt); p-adic analogs of ELLIPTIC FUNCTIONS [V.31] (with his student Elisabeth Lutz).

Starting in 1940, Weil became active on what was probably the biggest challenge in arithmetic algebraic geometry at the time. Helmut Hasse had proved in 1932 the analogue of the RIEMANN HYPOTHESIS [IV.2 §3] for curves of genus 1 (elliptic curves) defined over a field with finitely many elements. The problem was to generalize this to algebraic curves of genus higher than 1. In 1936, Max Deuring had proposed algebraic correspondences as a crucial new ingredient for attacking this problem; but the problem remained open until World War II. Weil’s initial attempt, written while in jail in Rouen, was very modest, and contained little more than Deuring’s observations of 1936. But, after several years of searching in various directions while in residence in the United States, Weil finally became the first person to prove the analogue of the Riemann hypothesis for all nonsingular curves. This proof relied on his complete rewriting of algebraic geometry (over an arbitrary ground field), which he had published before in his Foundations of Algebraic Geometry (1946). Furthermore, Weil generalized the analogue of the Riemann hypothesis from curves to algebraic varieties of arbitrary dimensions, defined over a finite field, and added a new topological interpretation of the main invariants of the relevant zeta functions. Taken together, all these conjectures became known as THE WEIL CONJECTURES [V.35]; they represented the most important stimulus for the further development of algebraic geometry right through to the 1970s, and to some extent later as well.

Several mathematicians were at work in the 1930s and 1940s trying to rewrite algebraic geometry. Weil’s Foundations, even though it does contain striking new insights (e.g., a novel definition of intersection multiplicity), owes its basic notions (generic points, specializations) to van der Waerden, and it exerted its influence on the mathematical community in conjunction with the (different) rewriting of algebraic geometry developed so successfully by Oscar Zariski from 1938 onward. It was therefore to a large extent the characteristic style, rather than just the “mathematical content,” of the Foundations that would create a new way of doing algebraic geometry for the next twenty years or so, until it began to be replaced by Grothendieck’s language of schemes.

Later works include, among other seminal papers and books, Weil’s “adelic” rewriting of Siegel’s work on quadratic forms, and a crucial contribution to the philosophy, due to Taniyama and Shimura, that elliptic curves over the rational numbers should be modular—the proof of this fact is the basis of Wiles’s 1995 proof of FERMAT’S LAST THEOREM [V.10].

In 1947, Weil—whose evasion of the French draft in 1939 was considered very critically by many American colleagues—finally obtained a professorship at a distinguished university, namely Chicago. In 1958, he moved to Princeton as a permanent member of the Institute for Advanced Study.

The postwar years saw Weil continuously active on many fronts of mathematical research, contributing insightfully to many subjects that were in the air at the time. To mention just a few: the Weil groups of CLASS FIELD THEORY [V.28]; the explicit formulas of analytic number theory; various aspects of differential geometry, in particular KÄHLER MANIFOLDS [III.88 §3]; the determination of Dirichlet series by their functional equations. All of these topics point to seminal works without which today’s mathematics would not be what it is.

In his later years, Weil put his erudition and historical sense to work writing articles and a book on the history of mathematics: Number Theory, an Approach through History. He also published a partial autobiography ending in 1945, Souvenirs d’Apprentissage, of considerable literary quality.

Further Reading

Weil, A. 1976. Elliptic Functions According to Eisenstein and Kronecker. Ergebnisse der Mathematik und ihrer Grenzgebiete, volume 88. Berlin: Springer.

——. 1980. Oeuvres Scientifiques/Collected Papers, second edn. Berlin: Springer.

——. 1984. Number Theory. An Approach through History. From Hammurapi to Legendre. Boston, MA: Birkhäuser.

——. 1991. Souvenirs d’Apprentissage. Basel: Birkhäuser 1991. (English translation: 1992, The Apprenticeship of a Mathematician. Basel: Birkhäuser.)

Norbert Schappacher and Birgit Petri