NUMBER-THEORY AND ALGEBRAIC GEOMETRY

ANDRE WEIL

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 Kro-necker. 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 Xl th 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,

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NUMBER-THEORY AND ALGEBRAIC GEOMETRY 91

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 <r(X) = x, <r(Y) = y, and cr(a) = a for every a £ k; this homomorphism is also called a specialization of that ring, and a generic one if it is an isomorphism of it into kf ; consequently, (x, y) will be called a specialization of (X, Y), and will be called generic if cr is generic.

Our homomorphism a has been so defined as to preserve the elements of the "ground-field" k; but this restriction, usual as it is in algebraic geometry, may well prove too narrow for some purposes. If, for example, we consider a curve F(X, Y, t) = 0, depending upon a parameter t, where F is a polynomial in X, Y, i with coefficients in a field k, then the coefficients of the equation of the curve are in the field k(t). However, with our curve, we naturally associate the surface F(X, Y, T) = 0; the curve then appears as a plane section of that sur­face by the plane T = t. Because of this changed point of view, the parameter t, previously frozen by its inclusion in the field of "constants", is now liberated and available for specialization; and so we are free now to consider as a spe­cialization of our ring k[X, Y, t]/(F(X, Y, t)) any homomorphism of that ring into an extension k' of k, still preserving the elements of k, but mapping X = X mod F,Y = Y mod F, t = t mod F onto any three elements xr, yf, t' of k' satis­fying F(xf, y', tf) = 0. Thus no longer restricted to the exclusive consideration of the "generic" curve belonging to the family F(X, Y, t) = 0, we are enabled to consider any specialization F(X, Y, t') = 0 of that curve, and the whole surface F(X, Y, t) = 0 spanned by that family.

This shifting of our point of view necessitates a re-examination of the concept of ground-field and of the field of definition of a variety. The previous speaker has mentioned, as one of the main achievements of modern algebraic geometry, the possibility of operating over quite arbitrary ground-fields. One should not be blind, however, to the somewhat illusory nature of this achievement. As our knowledge of algebraic curves is fairly extensive, there is, it is true, a great deal that we can say on the curve F(X, Y, t) = 0 depending upon the parameter t, in the example discussed above; and we should not possess that knowledge if our methods of proof were not valid over the ground-field k(t). But as we have pointed out, all we can say on the curve F(X, Y, t) = 0 is but part of the the­ory of the surface F(X, Y, T) = 0. This may be at the present moment, and it is in fact, one of the best ways of acquiring some knowledge of the geometry on that surface; but the fact remains that, in the final analysis, any statement on a variety with a larger ground-field boils down to a statement on a variety (of higher dimension, and therefore intrinsically more difficult to study) over a smaller ground-field.

Now consider, with Kronecker, that, in most problems of algebraic geometry, only a finite number of points and varieties occur at a time ; these will necessarily have a common field of definition which is finitely generated over the prime

92 ANDRÉ WEIL

field, i.e., which is generated over the prime field (the field Q of rational num­bers if the characteristic is 0, and otherwise a finite field) by a finite number of quantities (h, • • • , tN) ; if these are considered as parameters, and are made available for specialization, then, in the final analysis, every statement we can make can be thought of as a theorem in algebraic geometry over an absolutely alge­braic ground-field, i.e., either over a finite field or over ßn algebraic number-field of finite degree. While this realization, of course, cannot in any way detract from the methodological importance of arbitrary ground-fields as one of the chief tools of modern algebraic geometers, it gives us some insight into the deep meaning of Kronecker's view, according to which the absolutely algebraic fields are the natural ground-fields of algebraic geometry, at any rate as long as purely algebraic methods (as distinct from analytical or topological methods) are being used. Now these are fields with strongly marked individual features, which will undoubtedly have to be taken more and more into account as alge­braic geometry develops along more Kroneckerian lines. For instance, the field with q elements can be characterized by the fact that its elements are invariant under the automorphism x —> 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 <r of the ring R[X, Y]/(F) into a field kf will be called a specialization of that ring; if in particular it pre­serves (or at least if it maps isomorphically) the elements of R, then it can be extended to a homomorphism of k[X, Y]/(F) which preserves the elements of k.

As Kronecker realized, this affects our concept of dimension. Take for in­stance, instead of our curve, a hypersurface F(X\, • • • , Xn) = 0 in n-dimensional space, with coefficients in a subring R of a field k; let 9t denote the ring R [Xi, • • • , Xn]/(F). Then the dimension n — 1 of that variety can be defined as the degree of transcendency, over the ground-field k, of the function-field on the variety, i.e., of the field of fractions of 9$, or, equivalently, as the maximum number of successive specializations <r, af, a", • • • , of 9? onto a ring 9Î', of 9t' onto a ring 9î", etc., each one of which preserves the elements of R, and none of which is an isomorphism; the rings 9Î', 9î", • • • are understood to be "integral domains" (i.e., subrings of fields). If we remove the condition that the speciali­zations <r must preserve the elements of R, but merely require that they should preserve the elements of the "prime ring" (the ring Z of integers if the charac­teristic is 0, the ring of integers mod p if it is p > 1), this gives us the dimension over the prime field, or absolute dimension. So far, we have not crossed the boundaries of ordinary algebraic geometry, even though we may have pushed down the ground-field to an absolutely algebraic field. In particular, if the char-

NUMBER-THEORY AND ALGEBRAIC GEOMETRY 93

acteristic is p > 1, every homomorphism must preserve the elements of the prime field, and so there is no temptation, nor even any possibility, for us to cross those boundaries. However, if the characteristic is 0, there are homo­morphisms which do not preserve the characteristic; as soon as we allow these to enter the picture, we are within a wider area, where algebraic geometry and number-theory commingle and cannot be kept apart; and, as a consequence, the proper concept of dimension is the Kroneckerian concept. Since our se­quences of specializations <r, af, • • • can now be increased by one which changes the characteristic from 0 to some p > 1, it follows that the Kroneckerian dimen­sion is higher by 1 than that of algebraic geometry proper. For instance, a curve over an algebraic number-field has the Kroneckerian dimension 2,

In this sense, the only two cases of dimension 1 are those of a curve over a finite field, and of an algebraic number-field. In fact, it has been well known, ever since Kronecker and Dedekind, that there are far-reaching analogies be­tween these two cases, and these have been among the chief sources of progress in both directions; indeed, we have reached a stage where we can deal simul­taneously with large segments of both theories, not merely the more elementary ones, but also class-field theory and part of the theory of the zeta-function. It is true that these analogies are still incomplete at some crucial points; new con­cepts are clearly needed before we can transport to number-fields, even con-jecturally, the facts about the Jacobian variety of a curve which have recently led to the proof of the Riemann hypothesis ([lie], [llf]). Nevertheless, our knowl­edge of these topics is fairly extensive, whereas the same can hardly be said of the problems in higher dimensions.

I t is true that the theory of local rings has been extensively developed, largely by its initiator Krull (cf. e.g. [6]), and more recently by Chevalley ([2]), I. Cohen ([4]), and others. Such rings arise as follows: <r being, as above, a spe­cialization, say, of the ring 9Î = R[X, Y]/(F) defined by a curve F(X, Y) = 0, it can be extended to a homomorphic mapping of the ring 9Î' of those elements u/v of the field of fractions $ of 9Î, for which u, v are in 9? and ov ^ 0, by put­ting c(u/v) = au/av', 3Î' is the specialization-ring, and the ideal of non-units in 9Î', which is the kernel of <r, is the specialization-ideal; dlf is called a local ring, and its completion, with respect to the topology defined on it by the powers of the specialization-ideal, is a complete local ring; experience shows that it is desirable to confine oneself to integrally closed specialization-rings, and this leads to Zariski's fundamental concept of normality. Up to now geometers have used only characteristic-preserving specializations; therefore all their local rings contain a field, and have the same characteristic as their residue-class ring. Fortunately algebraists have not confined themselves to that case, so that their work is immediately available for the more general geometry that we are en­visaging here.

We are thus led to modify the Kroneckerian view that the "true" or "natural" ground-fields in algebraic geometry are the absolutely algebraic fields; this is so as long as ground-fields are considered from the purely algebraic point of view,

94 ANDRÉ WEIL

without any additional structure. However, it is now clear that the study of a family of varieties at, or rather in the neighborhood of, a given specialization of the parameter leads at once to the consideration of algebraic varieties over complete local rings and their fields of fractions; some recent work by Chow ([3]) may be considered as pertaining to this subject, of which the "geometry on a variety in the neighborhood of a subvariety" (as exemplified chiefly by Zariski's theory of holomorphic functions ([12]) forms a natural extension. That this does not contradict the Kroneckerian outlook, but has its root in it, is clearly shown by the fact that the theory of local rings was originated by Hensel; his p-adic rings, in fact, are the complete local rings attached to the specializa­tions of the rings of integers in algebraic number-fields. Hence the local study, say, of an algebraic curve F(X, Y) = 0 with coefficients in the ring Z of rational integers, "at" the specialization of Z determined by a prime p, amounts to en­larging the ground-field to the p-adic field. Thus the p-adic fields appear as another kind of "natural" ground-fieTd, and one may expect that the geometry over such fields will acquire more and more importance as it learns to develop its own methods. One may quote here E. Lutz's results on elliptic curves ([7]), showing that the group of points on such a curve has a subgroup of finite index, isomorphic to the additive group of integers in the ground-field; similar results undoubtedly hold for Abelian varieties of any dimension. In his beautiful thesis, Chabauty ([1]), following ideas of Skolem (LIO]), has shown how the method of p-adic completion, with respect to a more or less arbitrary prime p, can yield deep results about varieties over an algebraic number-field; there, as already in Skolem's work, the problem concerns the intersection of an algebraic variety and of a multiplicative group; by p-adic completion, the latter becomes an algebroid variety defined by linear differential equations. Of course geometry over finite fields may in a certain sense be obtained from the geometry over p-adic fields by reduction modulo p, so that the latter may be said to contain all that the former contains, and a good deal more; but little use has been made so far of the relations between these two kinds of geometries, .and little is known about them.

But the geometry over p-adic fields, and more generally over complete local rings, can provide us only with local data; and the main tasks of algebraic geometry have always been understood to be of a global nature. It is well known that there can be no global theory of algebraic varieties unless one makes them "complete", by adding to them suitable "points at infinity," embedding them, for example, in projective spaces. In the theory of curves, for instance, one would not otherwise obtain such basic facts as that the numbers of poles and of zeros of a function are equal, or that the sum of residues of a differential is 0. One way of doing this (which, however, is effective only in the case of dimen­sion 1) consists in considering the valuations of the field of functions on the curve; on a given affine model F(X, Y) = 0, each simple point defines a valua­tion, viz., that one which assigns, to each function on the curve, the order of the pole or zero it may have at that point; and all valuations, with a finite num-

NUMBER-THEORY AND ALGEBRAIC GEOMETRY 95

ber of exceptions, can be so obtained; the exceptions correspond to the "multiple points" and to the "points at infinity", and give an invariant definition for these. Correspondingly, if we apply this idea to an algebraic number-field (also a one-dimensional problem), we obtain satisfactory formulations for global the­orems, entirely analogous to the theorems on algebraic curves, provided we allow for "archimedean" valuations with somewhat weaker properties than those of algebraic geometry and than the p-adic valuations on number fields, viz., those for which the completed field is the field of real or that of complex numbers. Thus it appears that algebraic geometry over the complex number-field is, after all, a legitimate object of study, no less necessary or useful than geometry over p-adic fields; and so the door is opened to topology, function-theory, differential geometry, and partial differential equations. This, at any rate, is the logical way in which algebraic geometry over complex numbers ought to have been bom, had mathematics consisted solely of number-theory and algebra. That it came into being quite differently, and that it developed so far ahead of other branches of geometry, is a historical accident; it is indeed a fortunate one, having allowed free play to a tool which is invaluable as long as one is aware of its limitations; I need hardly tell you that I am referring to our spatial intuition.

We are now ready to consider in more specific terms the few known results in the "geometry over integers", which, following Kronecker, I have been trying to define ; and for this we must turn first of all, naturally, to Kronecker himself. His great work on elliptic functions ([5b]), or rather its algebraic part (as dis­tinct from the equally profound analytical theory), gives us a first example of an investigation of that kind; this consists in the study of the equation Y2 = 1 — pX2 + Xi over the ring Z\p], where p is an indeterminate, and is chiefly concerned with the transformation of elliptic functions. Jacobi's results on this subject are interpreted as defining, for every odd prime p, a correspondence between two generic points (x, y, p), (x', yf, pf) of the surface Y2 = 1 — pX2 + X4, where %' = xnF(l/x)/F(x), y' = G(x)/F(x)2, p' is algebraic over Q(p), and F, G are polynomials with algebraic coefficients over Q(p). Let a- be a root of 1 - pX2 + X4 = 0, and o-' a root of 1 - pfX2 + Xe = 0. Then Kronecker proves the following facts. The coefficients of F, G are in the field Q(cr, <rf); if divisibility relations are understood in the sense of integral algebraic elements over Z[p], then </ and all the roots of G(X) are units; and F(X) is of the form

F(X) « IT X*"1 + ir Ë ytX*"'-1 + 1,

where w and the yi are integral over Z[p] ; furthermore, ir is of degree p + 1 over Q(p), and has the norm p over that field. The main results on complex multipli­cation, and its application to the class-field theory of imaginary quadratic fields, can be derived from these facts by specialization of the parameter p. I t is very probable that a reconsideration of this splendid work from a modern point of view would not merely enrich our knowledge of elliptic function-fields, but

96 ANDRÉ WEIL

would also reveal principles of great importance for any further development of algebraic geometry over integers.

Now, coming back to the Grundzüge,-take Kronecker's well-known and sup­posedly outmoded device for the introduction of ideals. This consists in asso­ciating with the elements OQ , a\, • • • , a^ of a ririg the linear expression Go + ]C?=i ufoi, or, when the homogeneous notation happens to be more suit­able, the linear form X/£=o ^ A > in the indeterminates Ui ; thus the Ui are new variables adjoined to the ring, a feature which, in the eyes of orthodox Dede-kindians, is a fatal blemish of this procedure. If, for instance, the a,- are in the ring k[X, Y]/(F) determined by a plane curve F(X, Y) = 0 with coefficients in a field k, the ideal generated by them means substantially the same as the set of common zeros of the ai, counted with their multiplicities; and this is again nothing else than the "fixed part" of the linear series cut by the variable linear variety 2™=o UiXi = 0 through the point 0, in the affine space of dimension m + 1, on the model of the given curve which is the locus of the point (aQ, • • • , am). If we translate this into the projective language, we find ourselves at the heart of the theory of linear series; and a slight extension of Kronecker's idea could lead us very naturally to such thoroughly "modern" topics as, for example, the associated form of a variety in projective space (the "Chow coor­dinates"). There is thus every reason to believe that the same idea will reacquire its full meaning in number-theory as soon as the interpénétration of number-theory and algebraic geometry, which Kronecker sought to realise, has been accomplished. Let us for instance try to define for number-fields a concept cor­responding to the degree of a projective curve. If / 0 , / i , • - • , fm are the coor­dinates of a generic point of a curve, and the Ui are indeterminates, the degree is the number of "variable" zeros of ^2iUifi(x); this must be equal to the num­ber of fixed poles minus the number of fixed zeros; in other words, if at every point P of the curve we put n(P) = mU<aP(fi), where œP(f) indicates the order of / at P , then the degree of the curve is d = — ̂ Pn(P). If we replace the fi by numbers & belonging to an extension k of degree n of the rational number-field Q and if ? is the point £ = (&>,•••,£«) in the projective ra-space, we are thus led to consider the number H(£) = YLv sup^(&), where the product is taken over all absolute values (p-adic or archimedean) of k; H(£) does hot change if the & are replaced by p&, with- p £ k. This concept is essentially due to Siegel [9]1; as D. G. Northcott indicates [8a], it is more convenient, for arithmetical purposes, to introduce the number h(%) = H(£)Un, which depends only upon the point £ and not upon the field k* We khall call h(£) the height of the point £. Following Kronecker, we may associate with the point £, with coordinates in k, the form F(u) = r-Nk/Q(%2 ^ù, where the Ui are indeterminates, and the rational number r is so chosen that the coefficients of F(u) are rational integers without common divisor. Then we have F(u) <$C (A(Ö ,J}w*)n (which means

1 Cf. also H. Hasse, Monatshefte für Mathematik vol. 48 (1939) p. 205. Actually there is a slight discrepancy between Northcott's definition of H(£) and that of Siegel and Hasse; we follow the latter.

NUMBER-THEORY AND ALGEBRAIC GEOMETRY • 97

that every coefficient of F is at most equal in absolute value to the corresponding one in the right-hand side) : hence, if n0 and hQ are given, there is at most a finite number of points £ for which n < nQ, h(£) < hQ. This is Northcott's lemma ([8a]; cf. [llh]), which is at the bottom of the application of the "infinite de­scent" to elliptic curves, and, more generally, to Abelian varieties over algebraic number-fields ([lia]; cf. [8b] and [lie]).

The height of a variable point on a curve or on a variety can best be studied by means of the theory of distributions; this is the only chapter of Kroneckerian geometry which has been developed beyond the rudiments. Let us first consider a curve C, defined over an algebraic number-field k; if it is rational, i.e., if its function-field is the field k(t) generated over k by a single variable t, every func­tion f(t) on it can be written as f(t) — T]X'(£ ~ a%)mi> where 7 is a constant, the ai are the poles and zeros of the function, and the integers mi are their multi­plicities (counted positively for a zero, negatively for a pole). If the curve is not rational, such a representation is not possible, except in a merely symbolical manner, or else by means of transcendental multi-valued functions which cannot be used for arithmetical purposes. Let us, however, consider for a moment a definite embedding of k in the field of Complex numbers, so that C is defined over that field; and consider merely absolute values. Then one can attach to each point A of C a continuous real-valued function dA(M) on G, with 0 <! dA(M) < 1, which is 0 when M = A and only then, in such a way that if a function f(M) belonging to the function-field of G (over k or even over the field of complex numbers) has the zeros and poles A i with the multiplicities mi, then

f(M) = y(M)JldAi(M)mi, i

where y(M) is an inessential factor in the sense that there are constants 71, 72, both > 0, such that 71 < y(M) < y2 for all M. This can easily be verified by elementary topological methods. It can also be proved by an algebraic argu­ment, which remains valid if the field of complex numbers is replaced by the algebraic closure of the p-adic field, and also if the curve C is replaced by a variety. Reduced to its essential features, this argument can be described as follows. If y is a variety in an affine space, defined over the complex number-field or over a p-adic field, and if it does not contain the origin, then there is a polynomial P(Xi, • • • , Xn ) , vanishing on V and not at 0, with coefficients in the ground-field; this means that all points of V must satisfy an equation

1 — Y \ y ' i . . . y'n 1 — j^,aVl ... VnA.\ A. n ,

where all terms in the right-hand side are of degree > 1; therefore, if (xi, • • • , xn) is such a point, sup* | xi | cannot be arbitrarily small, and precisely it must be > 1 or > ( 2 I avi ••• vn | )_ 1 ; here | | denotes of course the ordinary orthep-adic absolute value, as the case may be.

So far we have considered only one absolute value, ordinary or p-adic, at a time, and so we have obtained, in this sense, merely "local" results; global re-

98 ANDRÉ WEIL

suits come from the consideration of all absolute values simultaneously; or else, what amounts to the same thing, one can treat the archimedean absolute values separately, in the manner indicated above, and then deal simultaneously with all the others. This is done by remarking that, if a variety V is defined, over an algebraic number-field k, and does not go through 0, its points must, as above, satisfy an equation

oto = JLfttvi ••• vn Xi • • • Xn

whose coefficients are algebraic integers in ft; and then if (xi, • • • , xn) is a point on V, with algebraic coordinated, the G.C.D. of the numerators of the frac­tional principal ideals (xi), • • • , (xn) must divide the principal ideal (a0) of ft. Out of this very simple fact one derives all the known results of the theory of distributions,- one of whose main results is the following "theorem of decompo­sition":

Let G be a curve, defined over an algebraic number-field ft. One can attach to each algebraic point A on G a function aA(M), defined at all algebraic points M of G, whose value at M is an ideal of the algebraic number-field ft (A, M), so that the following properties hold: aA(M) is 0 when M — A and only then; and whenever / is a function on G, having the zeros and poles Ai with the mul­tiplicities mi, then the principal fractional ideal (f(M)) has the expression

(/m) = ow ifouW1, where c(M) is an inessential fractional ideal in the sense that both c(M) and c(M)~~x divide a fixed natural integer. Furthermore, exactly the same result. holds for every nonsingular projective variety V of any dimension r, except that, of course, the ideal-valued functions a(M) are then attached, not to the points of V, but to the subvarieties of V of dimension r — 1.

As we have said above, this becomes a truly global result if we combine it with the corresponding result over complex numbers. When this is done, one finds inequalities for the height of a variable point on a, projective variety, which is found to depend essentially only upon the class of the divisors in the linear series determined on the variety by its hyperplane sections. In particular, let G be a curve of degree d in a projective space; let G' be a curve, birationally equivalent to G, of degree d', in the same or in another projective space; let M, Mf be corresponding points on G, C, with algebraic coordinates; then, to every e, there are constants 71, 72, both > 0, such that

yih(M)lld-e < h(M')lld' < y2h(M)lld+t

for all pairs of corresponding points M, Mf on Ç; in this sense, the "order of magnitude" of h(M)1,d is independent of the projective model chosen for C. This is the decisive inequality for Siegel's proof of the fact that a nonrational curve can have at most a finite number of points with integral coordinates in a given algebraic number-field ([9]; cf. [Uh]). The same approach also leads very simply to Northcott's inequalities ([8]; cf. [Uh]); these contain as special cases

NUMBER-THEORY AND ALGEBRAIC GEOMETRY 99

the inequalities by which it was first proved that the points on an Abelian vari­ety, with coordinates in a given algebraic number-field, form a finitely generated group ([lia]; cf. [lie]), so that a thoroughly "modernized" version of that proof could now be given.

I should like to conclude with a brief discussion of a very interesting conjec­ture, due, I believe, to Hasse. As we have said, from the Kroneckerian point of view the fields of dimension 1 are the number-fields and the function-fields of curves over finite fields; to each one of these there belongs a zeta-function, the properties of which may be said to epitomize in analytic garb some of the more important properties of the field. It is therefore reasonable to guess that similar functions can be attached to fields of higher dimension, and in the first place to the fields of dimension 2, i.e., to the curves over an algebraic number-field, and to the surfaces over a finite field. Consider the latter problem first: let S be a surface over the finite field ft of q elements; and define NP, for each v, as the number of points on the surface with coordinates in the extension kv of degree v of the ground-field; the analogy with curves, as well as the consideration of some special cases, makes it very natural ([llg]) to introduce the function Z(q~°), where Z(U) is defined by Z(0) = 1, d log Z(U)/dU = J^iN^U^1, and to expect that this will have the essential properties of a zeta-function over a finite field; i.e., that it is a rational function of U, that it satisfies a functional equation, and that it satisfies a suitably modified Riemann hypothesis; even the first property seems exceedingly difficult to prove at present, except in special cases. Now, suppose that we have on S a family of curves C(t) depending upon a parameter t; for simplicity assume that C(t) depends rationally upon t, and that no two curves C(t) have a point in common. If we give to t a value which is algebraic over ft, C(i) will be defined over k(t), and a zeta-function will be attached to it, defined in a manner similar to that employed for S. As the number of points on S with coordinates in ft„ is obviously the sum of the same numbers for all the curves C(t), it follows at once that Z(U) is the product of the zeta-f unctions attached to the curves C(t), provided that we take only one representative for each set of curves conjugate to each other over ft. Now this definition may at once be transported to number-fields: if G is a curve over the algebraic number-field K, given by an equation F(X, Y) = 0, then, for almost all prime ideals p of K, the equation F = 0, reduced modulo p, will define a curve of the same genus as G over the finite field of q = N($) elements; this has a zeta-function; and we are thus led to consider the product of these zeta-functions for all p, which is precisely the function previously defined by Hasse, of which he conjectured that it can be continued analytically over the whole plane, that it is meromorphic, and that it satisfies a functional equation. In a few simple cases, this function can actually be computed; e.g., for the curve Y2 = X* — 1 it can be expressed in terms of Hecke's L-functions for the field ft(\/l) ; this example also shows that such functions have infinitely many poles, which is a clear indication of the very considerable difficulties that one may expect in their study.

100 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.

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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, Abhand­

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UNIVERSITY OF CHICAGO,

CHICAGO, I I I . , IT. S. A.

INTERNATIONAL CONGRESS

OF

MATHEMATICIANS

Cambridge, Massachusetts, U. S. A.

1950

CONFERENCE IN ANALYSIS

Committee

Marston Morse (Chairman)

L. V. Ahlfors G. C. Evans

Salomon Bochner EINAR HILLB