NUMBER-THEORY AND ALGEBRAIC GEOMETRY ANDRE W E I L

Mr. Chairman, Ladies and Gentlemen, The previous speaker concluded his address with a reference to Dedekind and Weber. It is therefore fitting that I should begin with a homage to Kronecker. There appears to have been a certain feeling of rivalry, both scientific and personal, between Dedekind and Kronecker during their life-time; this developed into a feud between their followers, which was carried on until the partisans of Dedekind, fighting under the banner of the "purity of algebra", seemed to have won the field, and to have exterminated or converted their foes. Thus many of Kronecker's far-reaching ideas and fruitful results now lie buried in the impressive but seldom opened volumes of his Complete Works. While each line of Dedekind's X l t h Supplement, in its three successive and increasingly "pure" versions, has been scanned and analyzed, axiomatized and generalized, Kronecker's once famous Grundzüge are either forgotten, or are thought of merely as presenting an inferior (and less pure) method for achieving part of the same results, viz., the foundation of ideal-theory and of the theory of algebraic number-fields. In more recent years, it is true, the fashion has veered to a more multiplicative and less additive approach than Dedekind's, to an emphasis on valuations rather than ideals; but, while this trend has taken us back to Kronecker's most faithful disciple, Hensel, it has stopped short of the master himself. Now it is time for us to realize that, in his Grundzüge, Kronecker did not merely intend to give his own treatment of the basic problems of ideal-theory which form the main subject of Dedekind's life-work. His aim was a higher one. He was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number-theory and algebraic geometry as special cases. This grandiose conception has been allowed to fade out of our sight, partly because of the intrinsic difficulties of carrying it out, partly owing to historical accidents and to the temporary successes of the partisans of purity and of Dedekind. It will be the main purpose of this lecture to try to rescue it from oblivion, to revive it, and to describe the few modern results which may be considered as belonging to the Kroneckerian program. Let us start from the concept of a point on a variety, or, what amounts to much the same thing, of a specialization. Take for instance a plane curve G, defined by an irreducible equation F(X, Y) = 0, with coefficients in a field k. A point of C is a solution (x, y) of F(X, Y) = 0, consisting of elements x] y of some field k' containing k. In order to define the function-field on the curve, we identify two polynomials in X, Y if they differ only by a multiple of F, i.e., we build the ring k[X, Y]/(F), and we take the field of fractions $ of that ring: in particular, X and Y themselves determine the elements X = X mod F, 90

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Y = Y mod F, of $, and (X, Y) is a point of C, called generic since it does not satisfy any relation over k except F(X, Y) = 0 and its consequences. Then any point (x, y) of C, with coordinates in an extension kf of k, determines a homomorphism cr of the ring k[X, Y]/(F) into/c', d efined by putting xq of any field containing it; this must have a profound influence on the geometry over that field; and recent work connected with the Riemann hypothesis ([lie]) fully confirms that expectation. Another fact, so far an isolated one, in the same direction, is the existence of matrices, associated with curves over a finite field, which bear a curious resemblance with the period-matrices of abelian integrals in the classical theory (cf. [lid]). We are now in a position to discussspecializations again from our broadened point of view. If e.g. F(X, Y) = 0 is the equation of a curve, with coefficients in a subring R of a field k, then any homomorphism
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ANDRÉ WEIL REFERENCES

l . C . CHABATJTY, Sur les équations diophantiennes liées aux unités d'un corps de nombres algébriques fini, Annali di Matematica (IV) vol. 17 (1938) p. 127. 2. C. CHEVALLEY, On the theory of local rings, Ann. of Math. vol. 44 (1943) p. 690. 3. W. L. CHOW, Algebraic systems of positive cycles in an algebraic variety, Amer. J. Math. vol. 72 (1950) p. 247. 4. I. S. COHEN, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. vol. 59 (1946) p. 54. 5. L. KRONECKER, a) Grundzuge einer arithmetischen Theorie der algebraischen Grössen, Werke, vol. II, Leipzig, Teubner, 1897, pp. 237-387. b) Zur- Theorie der elliptischen Funktionen, Werke, vol. IV, Leipzig-Berlin, Teubner, 1929, pp. 345-495. 6. W. KRULL, Dimensionstheorie in Stellenringen, J. Reine Angew. Math. vol. 179 (1938) p. 204. 7. E. LUTZ, Sur Véquation y2 = xz — Ax — B dans les corps ^-adiques, J. Reine Angew. Math. vol. 177 (1937) p. 238. 8. D. G. NORTHCOTT, a) An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. vol. 45 (1949) p. 502. b) A further inequality in the theory of arithmetic on algebraic varieties, ibid. p. 510. 9. C. L. SIEGEL, Über einige Anwendungen diophantischer Approximationen, Abhandlungen der Preussischen Akademie der Wissenschaften (Jahrgang 1929, no. 1), Berlin, 1930. 10. TH. SKOLEM, Einige Sätze über ip-adische Potenzreihen mit Anwendung auf gewisse exponentielle Gleichungen, Math. Ann. vol. 111 (1935) p. 399. 11. A. WEIL, a) L'arithmétique sur les courbes algébriques, Acta Math. vol. 52 (1928) p. 281 (Thèse, Paris 1928). b) Arithmétique et géométrie sur les variétés algébriques, Actualités Scientifiques et Industrielles, no. 206 (Exp. math, à la mém. de J. Herbrand XI), Paris, Hermann et Cie, 1935. c) Arithmetic of algebraic varieties (Russian), Uspehi Matematieeskih Nauk, vol. 3 (1937) p. 101. d) Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci. Paris vol. 210 (1940) p. 592. e) Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Scientifiques et Industrielles, no. 1041 (Pubi. Inst. Math. Strasb., VII), Paris, Hermann et Cie, 1948. f) Variétés abêliennes et courbes algébriques, Actualités Scientifiques et Industrielles, no. 1064 (Pubi. Inst. Math. Strasb. VIII), Paris, Hermann et Cie, 1948. g) Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. vol. 55 (1949) p. 497. h) Arithmetic on algebraic varieties, Ann. of Math. vol. 53 (1951) pp. 412-444, 12. O. ZARISKI, Theory and applications of holomorphic functions on algebraic varieties, Memoirs of Amer. Math. Soc, no. 5 (1951). UNIVERSITY OF CHICAGO, C H I C A G O , I I I . , IT. S.

A.

INTERNATIONAL CONGRESS OF

MATHEMATICIANS

Cambridge, Massachusetts, U. S. A. 1950

CONFERENCE I N ANALYSIS

Committee Marston Morse (Chairman) L. V. Ahlfors

G. C. Evans

Salomon Bochner

EINAR HILLB