The Riemann Hypothesis Over Finite Fields: from Weil to the Present Day* by James S

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The Riemann Hypothesis over Finite Fields: From Weil to the Present Day* by James S. Milne†

Abstract. The statement of the Riemann hypoth- def 1 ζ(K,s) = ∏ −s esis makes sense for all global fields, not just p 1 − Np the rational numbers. For function fields, it has a where p runs over the prime ideals of the integral clo- natural restatement in terms of the associated curve. sure O of k[x] in K and Np = |O /p|; we also include Weil’s work on the Riemann hypothesis for curves K K factors for each of the finitely many prime ideals p in over finite fields led him to state his famous “Weil the integral closure of k[x−1] not corresponding to an conjectures”, which drove much of the progress in ideal of O . The Riemann hypothesis for K states that algebraic and arithmetic geometry in the following K the zeros of ζ(K,s) lie on the line ℜ(s) = 1/2. decades. Let C be the (unique) nonsingular projective over MSC 2010 subject classifications: 01A60, 11-03, k with function field is K. Let kn denote the finite field 14-03. of degree n over k, and let Nn denote the number of points of C rational over kn:

N def= |C(k )|. Contents n n The zeta function of C is the power series Z(C,T) ∈ 1 Weil’s Work in the 1940s and 1950s ...... 15 Q[[T]] such that 2 Weil Cohomology ...... 27 ∞ 3 The Standard Conjectures ...... 34 d logZ(C,T) n−1 = ∑ NnT . 4 Motives ...... 38 dT n=1 5 Deligne’s Proof of the Riemann Hypothesis over Finite Fields ...... 41 This can also be expressed as a product 6 The Hasse-Weil Zeta Function ...... 46 1 Z(C,T) = ∏ deg(P) P 1 − T A global field K of characteristic p is a field finitely generated and of transcendence degree 1 where P runs over the closed points of C (as a scheme). over Fp. The field of constants k of K is the algebraic Each P corresponds to a prime ideal p in the preceding deg(P) closure of Fp in K. Let x be an element of K transcen- paragraph, and Np = q where q = |k|. Therefore dental over k. Then ζ(K,s) = Z(C,q−s). * Reprinted with permission from “The legacy of Bernhard Riemann after one hundred and fifty years”, edited by Lizhen Let g be the genus of C. Using the Riemann-Roch Ji, Frans Oort and Shing-Tung Yau, International Press, 2016. theorem, one finds that † Professor Emeritus, Department of Mathematics, Univer- sity of Michigan, Ann Arbor, Mich., U.S.A. P(C,T) Z(C,T) = E-mail: [email protected] (1 − T)(1 − qT)

14 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 where The Proof of the Riemann Hypothesis for Elliptic Curves 2g For future reference, we sketch the proof of the P(C,T) = 1 + c1T + ··· + c2gT ∈ Z[T]; Riemann hypothesis for elliptic curves. Let E be such moreover Z(C,T) satisfies the functional equation a curve over a field k, and let α be an endomorphism def al of E. For ` 6= char(k), the Z`-module TÈ = lim E`n (k ) Z(C,1/qT) = q1−g · T 2−2g · Z(C,T). ←−n is free of rank 2 and α acts on it with determinant Thus ζ(K,s) is a rational function in q−s with simple degα. Over C this statement can be proved by writing E = /Λ and noting that T E ' ⊗ Λ. Over a field of poles at s = 0, 1 and zeros at a1,...,a2g where the ai are C ` Z` Z the inverse roots of P(C,T), i.e., nonzero characteristic the proof is more difficult—it will be proved in a more general setting later (1.21, 2g 1.26). P(C,T) = ∏(1 − aiT), 2 i=1 Now let α be an endomorphism of E, and let T + cT + d be its characteristic polynomial on TÈ: and it satisfies a functional equation relating ζ(K,s) 2 def and ζ(K,1 − s). The Riemann hypothesis now asserts T + cT + d = det(T − α|TÈ). that 1 Then, |ai| = q 2 , i = 1,...,2g.

Note that d = det(−α|T`E) = deg(α)

( ∞ T n 1 + c + d = det(idE −α|T`E) = deg(idE −α), ∑1 Nn n logZ(C,T) = (1−a T)···(1−a T) log 1 2g , (1−T)(1−qT) and so c and d are integers independent of `. For an integer n, let and so n n n 2 0 0 Nn = 1 + q − (a1 + ··· + a2g). (1) T + c T + d = det(T − nα|T`E).

Therefore, the Riemann hypothesis implies that Then c0 = nc and d0 = n2d. On substituting m for T in n n 1/2 (1), we find that |Nn − q − 1| ≤ 2g · (q ) ; 2 2 conversely, if this inequality holds for all n, then the m + cmn + dn = det(m − nα|T`E). Riemann hypothesis holds for C.1 The above is a brief summary of the results ob- The right hand side is the degree of the map m − nα, tained by the German school of number theorists which is always nonnegative, and so the discriminant 2 2 (Artin, F. K. Schmidt, Deuring, Hasse, ....) by the c − 4d ≤ 0, i.e., c ≤ 4d. mid-1930s. In the 1930s Hasse gave two proofs of the We apply these statements to the Frobenius q q Riemann hypothesis for curves of genus 1, the second map π : (x0 : x1 : ...) 7→ (x0 : x1 : ...). The homomorphism al of which uses the endomorphism ring of the ellip- idE −π : E → E is étale and its kernel on E(k ) is E(k), 2 tic curve2 in an essential way (see the sketch below). and so its degree is |E(k)|. Let f = T + cT + d be the Deuring recognized that for curves of higher genus characteristic polynomial of π. Then d = deg(π) = q, it was necessary to replace the endomorphism ring and c2 ≤ 4d = 4q. From of the curve with its ring of correspondences in the sense of Severi 1903, and he wrote two articles refor- |E(k)| = deg(idE −π) = det(idE −π|T`E) = f (1) = 1 + c + q, mulating part of Severi’s theory in terms of “double- we see that fields” (in particular, extending it to all characteristics). Weil was fully aware of these ideas of Deur- ||E(k)| − q − 1| = |c| ≤ 2q1/2, ing and Hasse. For an account of this early work, see Schappacher 2006 and Oort and Schappacher 2016. as required. 1 The condition implies that the power series 2g ∞ 2g ! 1 n n 1. Weil’s Work in the 1940s and 1950s ∑ = ∑ ∑ ai z i=1 1 − aiz n=0 i=1 −1/2 1/2 The 1940 and 1941 Announcements converges for |z| ≤ q , and so |ai| ≤ q for all i. The functional equation shows that q/ai is also a root of P(C,T), and Weil announced the proof of the Riemann hy- so |a | = q1/2. i pothesis for curves over finite fields in a brief three- 2 F. K. Schmidt (1931) showed that every curve over a finite field has a rational point. The choice of such a point on a page note (Weil 1940), which begins with the follow- curve of genus 1 makes it an elliptic curve. ing paragraph:

DECEMBER 2016 NOTICES OF THE ICCM 15 I shall summarize in this Note the solution of the main on C defines an endomorphism of G whose trace Weil problems in the theory of algebraic functions with a finite calls the trace Tr(X) of X. This is an element of ∏l6=p l, field of constants; we know that this theory has been the Z subject of numerous works, especially, in recent years, by which nowadays we prefer to define as the trace of X def Hasse and his students; as they have caught a glimpse of, on TG = lim G .7 ←−(n,p)=1 n the key to these problems is the theory of correspondences; After some preliminaries, including the definition but the algebraic theory of correspondences, due to Severi, is not sufficient, and it is necessary to extend Hurwitz’s tran- in this special case of what is now called the Weil scendental theory to these functions.3 pairing, Weil announces his “important lemma”: let 0 The “main problems” were the Riemann hypothesis X be an (m1,m2) correspondence on C, and let X be for curves of arbitrary genus, and Artin’s conjecture the (m2,m1)-correspondence obtained by reversing the that the (Artin) L-function attached to a nontrivial factors; simple Galois representation is entire. if m1 = g, we have in general (i.e., under conditions which 0 8 “Hurwitz’s transcendental theory” refers to the it is unnecessary to make precise here) 2m2 = Tr(X ◦ X ). memoir Hurwitz 1887, which Weil had studied al- From this he deduces that, for all correspondences X, ready as a student at the Ecole Normale Supérieure. the trace Tr(X ◦X0) is a rational integer ≥ 0 and that the There Hurwitz gave the first proof4 of the formula ex- number of coincident points of X is m +m −Tr(X). On pressing the number of coincident points of a corre- 1 2 applying these statements to the graph of the Frobe- spondence X on a complex algebraic curve C in terms nius map, he obtains his main results. of traces. In modern terms, At the time Weil wrote his note, he had lit- (number of coincidences) tle access to the mathematical literature. His confidence in the statements in his note was based on his = Tr(X|H0(C,Q)) − Tr(X|H1(C,Q)) + Tr(X|H2(C,Q)). own rather ad hoc heuristic calculations (Œuvres, I, This can be rewritten pp. 548–550). As he later wrote:

(2) (X · ∆) = d2(X) − Tr(X|H1(C)) + d1(X) In other circumstances, publication would have seemed very premature. But in April 1940, who could be sure of a where d (X) = (X ·C × pt) and d (X) = (X · pt ×C). tomorrow? It seemed to me that my ideas contained enough 1 2 substance to merit not being in danger of being lost.9 (Œu- Now consider a nonsingular projective curve C0 vres I, p. 550.) over a field k0 with q elements, and let C be the curve In his note, Weil had replaced the jacobian variety over the algebraic closure k of k0 obtained from C by extension of scalars. For the graph of the Frobenius with its points of finite order. But he soon realized endomorphism π of C, the equality (2) becomes that proving his “important lemma” depended above all on intersection theory on the jacobian variety. (Γπ · ∆) = 1 − “Tr(π|H1(C))” + q. In 1940 I had seen fit to replace the jacobian with its As (Γ · ∆) = |C (k )|, we see that everything comes group of points of finite order. But I began to realize that a π 0 0 considerable part of Italian geometry was based entirely on down to understanding algebraically (and in nonzero intersection theory… In particular, my “important lemma” characteristic!), the “trace of a correspondence on the of 1940 seemed to depend primarily on intersection theory first homology group of the curve”. on the jacobian; so this is what I needed.10 (Œuvres I, p. 556.) Let g be the genus of C . In his 1940 note,5 Weil 0 7 In terms of the jacobian variety J of C, which wasn’t avail- considers the group G of divisor classes on C of de- able to Weil at the time, TG = ∏l6=p Tl J, and the positivity of gree 0 and order prime to p, and assumes that G is Tr(X ◦X0) expresses the positivity of the Rosati involution on g 6 isomorphic to (Q/Z(non−p))2 . A correspondence X the endomorphism algebra of J. 8 Si m1 = g, on a en général (c’est-à-dire à des conditions qu’il 3 0 Je vais résumer dans cette Note la solution des principaux est inutile de préciser ici) 2m2 = Tr(X ◦ X ). problèmes de la théorie des fonctions algébriques à corps de 9 “En d’autres circonstances, une publication m’aurait paru constantes fini; on sait que celle-ci a fait l’objet de nombreux bien prématurée. Mais, en avril 1940, pouvait-on se croire as- travaux, et plus particulièrement, dans les derniéres années, suré du lendemain? Il me sembla que mes idées contenaient de ceux de Hasse et de ses élèves; comme ils l’ont entrevu, la assez de substance pour ne pas mériter d’être en danger de théorie des correspondances donne la clef de ces problèmes; se perdre.” This was seven months after Germany had in- mais la théorie algébrique des correspondances, qui est due vaded Poland, precipitating the Second World War. At the à Severi, n’y suffit point, et il faut étendre à ces fonctions la time, Weil was confined to a French military prison as a re- théorie transcendante de Hurwitz. sult of his failure to report for duty. His five-year prison 4 As Oort pointed out to me, Chasles-Cayley-Brill 1864, 1866, sentence was suspended when he joined a combat unit. Dur- 1873, 1874 discussed this topic before Hurwitz did, but per- ing the collapse of France, his unit was evacuated to Britain. haps not “in terms of traces”. Later he was repatriated to Vichy France, from where in Jan- 5 In fact, following the German tradition (Artin, Hasse, Deur- uary 1941 he managed to exit to the United States. Because ing, …), Weil expressed himself in this note in terms of func- of his family background, he was in particular danger during tions fields instead of curves. Later, he insisted on geometric this period. language. 10 En 1940 j’avais jugé opportun de substituer à la jaco- 6 See p. 19. bienne le groupe de ses points d’ordre fini. Mais je com-

16 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 In expanding the ideas in his note, he would construct Divisors the jacobian variety of a curve and develop a com- A divisor on V is a formal sum D = ∑niCi with ni ∈ prehensive theory of abelian varieties over arbitrary Z and C an irreducible curve on V. We say that D is posi- fields parallel to the transcendental theory over . i C tive, denoted D ≥ 0, if all the n ≥ 0. Every f ∈ k(V)× has But first he had to rewrite the foundations of alge- i an associated divisor ( f ) of zeros and poles—these braic geometry. are the principal divisors. Two divisors D and D0 are In the meantime, he had found a more elemen- said to be linearly equivalent if tary proof of the Riemann hypothesis, which involved only geometry on the product of two curves. First he D0 = D + ( f ) some f ∈ k(V)×. realized that if he used the fixed point formula (2) to define the trace σ(X) of a correspondence X on C, i.e., For a divisor D, let

def L(D) = { f ∈ k(V) | ( f ) + D ≥ }. σ(X) = d1(X) + d2(X) − (X · ∆), 0 then it was possible to prove directly that σ had many Then L(C) is a finite-dimensional vector space over k, of the properties expected of a trace on the ring of whose dimension we denote by l(D). The map g 7→ g f correspondence classes. Then, as he writes (Œuvres I, is an isomorphism L(D) → L(D − ( f )), and so l(D) de- p. 557): pends only on the linear equivalence class of D.

At the same time I returned seriously to the study of the Elementary Intersection Theory Trattato [Severi 1926]. The trace σ does not appear as such; but there is much talk of a “difetto di equivalenza” whose Because V is nonsingular, a curve C on V has a positivity is shown on page 254. I soon recognized it as my local equation at every closed point P of V, i.e., there integer σ(X ◦ X0) … I could see that, to ensure the validity of exists an f such that the Italian methods in characteristic p, all the foundations would have to be redone, but the work of van der Waerden, C = ( f ) + P. together with that of the topologists, allowed me to believe components not passing through that it would not be beyond my strength.11 If C and C0 are distinct irreducible curves on V, then Weil announced this proof in (Weil 1941). Before de- their intersection number at P ∈ C ∩C0 is scribing it, I present a simplified modern version of 0 def 0 the proof. (C ·C )P = dimk(OV,P/( f , f ))

Notes. Whereas the 1940 proof (implicitly) uses the where f and f 0 are local equations for C and C0 at P, jacobian J of the curve C, the 1941 proof involves and their (global) intersection number is only geometry on C × C. Both use the positivity of “σ(X ◦X0)” but the first proof realizes this integer as a 0 0 (C ·C ) = ∑ (C ·C )P. trace on the torsion group Jac(C)(non-p) whereas the P∈C∩C0 second expresses it in terms of intersection theory on C ×C. This definition extends by linearity to pairs of divisors D,D0 without common components. Now ob- serve that (( f ) ·C) = 0, because it equals the degree of The Geometric Proof of the Riemann Hypothesis the divisor of f |C on C, and so (D · D0) depends only for Curves on the linear equivalence classes of D and D0. This Let V be a nonsingular projective surface over an allows us to define (D · D0) for all pairs D,D0 by re- algebraically closed field k. placing D with a linearly equivalent divisor that intersects D0 properly. In particular, (D2) def= (D · D) is mençais à apercevoir qu’une notable partie de la géométrie italienne reposait exclusivement sur la théorie des intersec- defined. See Shafarevich 1994, IV, §1, for more de- tions. Les travaux de van der Waerden, bien qu’ils fussent tails. restés bien en deçà des besoins, donnaient lieu d’espérer que le tout pourrait un jour se transposer en caractéris- The Riemann-Roch Theorem tique p sans modification substantielle. En particulier, mon Recall that the Riemann-Roch theorem for a curve “lemme important” de 1940 semblait dépendre avant tout de la théorie des intersections sur la jacobienne; c’est donc C states that, for all divisors D on C, celle-ci qu’il me fallait. 11 En même temps je m’étais remis sérieusement à l’étude l(D) − l(KC − D) = deg(D) + 1 − g du Trattato. La trace σ a n’y apparaît pas en tant que telle; mais il y est fort question d’un “difetto di equivalenza” dont where g is the genus of C and KC is a canonical divisor la positivité est démontrée à la page 254. J’y reconnus bi- (so degKC = 2g − 2 and l(KC) = g). Better, in terms of 0 entôt mon entier σ(yy )… Je voyais bien que, pour s’assurer cohomology, de la validité des méthodes italiennes en caractéristique p, toutes les fondations seraient à reprendre, mais les travaux χ(O(D)) = deg(D) + χ(O) de van der Waerden, joints à ceux des topologues, donnaient 1 0 à croire que ce ne serait pas au dessus de mes forces. h (D) = h (KC − D).

DECEMBER 2016 NOTICES OF THE ICCM 17 The Riemann-Roch theorem for a surface V states by (3) above. Hence, suppose that (D · D) > 0. By the that, for all divisors D on V, Riemann-Roch theorem

1 (D · D) 2 l(D) − sup(D) + l(KV − D) = pa + 1 + (D · D − KV ) l(mD) + l(K − mD) ≥ m + lower powers of m. 2 V 2 where KV is a canonical divisor and Therefore, for a fixed m0 ≥ 1, we can find an m > 0 such that pa = χ(O) − 1 (arithmetic genus), sup(D) = superabundance of D(≥ 0,and = 0 for some divisors). l(mD) + l(KV − mD) ≥ m0 + 1 l(−mD) + l(K + mD) ≥ m + 1. Better, in terms of cohomology, V 0

1 If both l(mD) ≤ 1 and l(−mD) ≤ 1, then both l(KV − χ(O(D)) = χ(OV ) + (D · D − K) 2 mD) ≥ m0 and l(KV + mD) ≥ m0, and so h2(D) = h0(K − D), (4) l(2KV ) = l(KV − mD + KV + mD) ≥ l(KV + mD) ≥ m0. and so As m was arbitrary, this is impossible. 2 sup(D) = h1(D). 0 We shall also need the adjunction formula: let C Let Q be a symmetric bilinear form on a finite- be a curve on V; then dimensional vector space W over Q (or R). There ex- 2 2 ists a basis for W such that Q(x,x) = a1x1 + ··· + anxn. KC = (KV +C) ·C. The number of ai > 0 is called the index (of positivity) of Q—it is independent of the basis. There is the fol- The Hodge Index Theorem lowing (obvious) criterion: Q has index 1 if and only if Embed V in n. A hyperplane section of V is a di- there exists an x ∈ V such that Q(x,x) > 0 and Q(y,y) ≤ 0 P ⊥ visor of the form H = V ∩ H0 with H0 a hyperplane in for all y ∈ hxi . n Now consider a surface V as before, and let Pic(V) P not containing V. Any two hyperplane sections are linearly equivalent (obviously). denote the group of divisors on V modulo linear equivalence. We have a symmetric bi-additive inter- Lemma 1.1. For a divisor D and hyperplane section H, section form

(3) l(D) > 1 =⇒ (D · H) > 0. Pic(V) × Pic(V) → Z. On tensoring with and quotienting by the kernels, Proof. The hypothesis implies that there exists a D1 > Q 0 linearly equivalent to D. If the hyperplane H0 is cho- we get a nondegenerate intersection form sen not to contain a component of D1, then the hyper- 0 N(V) × N(V) → . plane section H = V ∩ H intersects D1 properly. Now Q D ∩H = D ∩H0, which is nonempty by dimension the- 1 1 Corollary 1.3. The intersection form on N(V) has in- ory, and so (D · H) > 0. 2 1 dex 1. Theorem 1.2 (Hodge Index Theorem). For a divisor Proof. Apply the theorem and the criterion just D and hyperplane section H, stated. 2 (D · H) = 0 =⇒ (D · D) ≤ 0. Corollary 1.4. Let D be a divisor on V such that (D2) > 0. If (D · D0) = 0, then (D02) ≤ 0. Proof. We begin with a remark: suppose that l(D) > 0, i.e., there exists an f 6= 0 such that ( f ) + D ≥ 0; then, Proof. The form is negative definite on hDi⊥. 2 for a divisor D0, The Inequality of Castelnuovo-Severi 0 0 0 (4) l(D + D ) = l((D + ( f )) + D ) ≥ l(D ). Now take V to be the product of two curves, V = C ×C . Identify C and C with the curves C × pt and We now prove the theorem. To prove the contra- 1 2 1 2 1 pt × C on V, and note that positive, it suffices to show that 2

(D · D) > 0 =⇒ l(mD) > 1 for some integer m, C1 ·C1 = 0 = C2 ·C2 C1 ·C2 = 1 = C2 ·C1. because then 1 Let D be a divisor on C1 × C2 and set d1 = D · C1 and (D · H) = (mD · H) 6= 0 d = D ·C . m 2 2

18 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Theorem 1.5 (Castelnuovo-Severi Inequality). Let D Aside 1.8. Note that, except for the last few lines, the be a divisor on V; then proof is purely geometric and takes place over an algebraically closed field.12 This is typical: study of the 2 (5) (D ) ≤ 2d1d2. Riemann hypothesis over finite fields suggests ques- Proof. We have tions in algebraic geometry whose resolution proves the hypothesis. 2 (C1 +C2) = 2 > 0 Correspondences (D − d2C1 − d1C2) · (C1 +C2) = 0. A divisor D on a product C1 ×C2 of curves is said Therefore, by the Hodge index theorem, to have valence zero if it is linearly equivalent to a

2 sum of divisors of the form C1 × pt and pt × C2. The (D − d2C1 − d1C2) ≤ 0. group of correspondences C (C1,C2) is the quotient of 2 On expanding this out, we find that D ≤ 2d1d2. 2 the group of divisors on C1 ×C2 by those of valence zero. When C1 = C2 = C, the composite of two divisors Define the equivalence defect (difetto di equiv- D and D is alenza) of a divisor D by 1 2 def ∗ ∗ 2 D1 ◦ D2 = p13∗(p12D1 · p23D2) def(D) = 2d1d2 − (D ) ≥ 0.

Corollary 1.6. Let D, D0 be divisors on V; then where the pi j are the projections C ×C ×C → C ×C; in general, it is only defined up to linear equivalence. 0 0 0 0 1/2 (6) D · D − d1d2 − d2d1 ≤ def(D)def(D ) . When D ◦ E is defined, we have Proof. Let m,n ∈ . On expanding out Z (8) d1(D ◦ E) = d1(D)d1(E), d2(D ◦ E) = d2(D)d2(E), 0 def(mD + nD0) ≥ 0, (D · E) = (D ◦ E ,∆) we find that where, as usual, E0 is obtained from E by reversing the factors. Composition makes the group C (C,C) of 2 0 0 0 2 0 m def(D) − 2mn (D · D ) − d1d2 − d2d1 + n def(D ) ≥ 0. correspondences on C into a ring R(C). Following Weil (cf. (2), p. 16), we define the “trace” As this holds for all m,n, it implies (6). 2 of a correspondence D on C by Example 1.7. Let f be a nonconstant morphism C1 → C2, and let gi denote the genus of Ci. The graph of f is σ(D) = d1(D) + d2(D) − (D · ∆). a divisor Γ f on C1 ×C2 with d2 = 1 and d1 equal to the degree of f . Now Applying (8), we find that

0 def 0 0 0 KΓ f = (KV + Γ f ) · Γ f (adjunction formula). σ(D ◦ D ) = d1(D ◦ D ) + d2(D ◦ D ) − ((D ◦ D ) · ∆) 2 = d1(D)d2(D) + d2(D)d1(D) − (D ) On using that KV = KC1 ×C2 +C1 × KC2 , and taking degrees, we find that = def(D).

2 0 2g1 − 2 = (Γ f ) + (2g1 − 2) · 1 + (2g2 − 2)deg( f ). Thus Weil’s inequality σ(D ◦ D ) ≥ 0 is a restatement of (5). Hence

(7) def(Γ f ) = 2g2 deg( f ). Sur les Courbes… (Weil 1948a)

Proof of the Riemann Hypothesis for Curves In this short book, Weil provides the details for his 1941 proof. He does not use the Riemann-Roch Let C be a projective nonsingular curve over a 0 theorem for a surface, which was not available in finite field k , and let C be the curve obtained by ex- 0 nonzero characteristic at the time. tension of scalars to the algebraic closure k of k . Let π 0 Let C be a nonsingular projective curve of genus g be the Frobenius endomorphism of C. Then (see (7)), over an algebraically closed field k. We assume a the- def(∆) = 2g and def(Γπ ) = 2gq, and so (see (6)), ory of intersections on C×C, for example, that in Weil 1/2 |(∆ · Γπ ) − q − 1| ≤ 2gq . 1946 or, more simply, that sketched above (p.17). We briefly sketch Weil’s proof that σ(D ◦ D0) ≥ 0. As we As 12 I once presented this proof in a lecture. At the end, a lis- (∆ · Γπ ) = number of points on C rational over k0, tener at the back triumphantly announced that I couldn’t have proved the Riemann hypothesis because I had only ever we obtain Riemann hypothesis for C0. worked over an algebraically closed field.

DECEMBER 2016 NOTICES OF THE ICCM 19 have just seen, this suffices to prove the Riemann hy- all nonzero ξ ∈ R(C) (ibid. Thm 10, p. 54). In the earlier pothesis. part of the book, Weil (re)proves the Riemann-Roch Assume initially that D is positive, that d2(D) = theorem for curves and develops the theory of corre- g ≥ 2, and that spondences on a curve based on the intersection theory developed in his Foundations. In the later part he D = D + ··· + D 1 g applies the inequality to obtain his results on the zeta with the Di distinct. We can regard D as a multivalued function of C, but he defers the proof of his results 13 map P 7→ D(P) = {D1(P),...,Dg(P)} from C to C. Then on Artin L-series to his second book. Some History D(k) = {(P,Di(P)) | P ∈ C(k), 1 ≤ i ≤ g} 0 D (k) = {(Di(P),P) | P ∈ C(k), 1 ≤ i ≤ g} Let C1 and C2 be two nonsingular projective curves 0 over an algebraically closed field k. Severi, in his fun- D ◦ D (k) = {(Di(P),D j(P)) | P ∈ C(k), 1 ≤ i, j ≤ g}. damental paper (1903), defined a bi-additive form

The points with i = j contribute a component Y1 = 0 σ(D,E) = d1(D)d2(E) + d2(D)d1(E) − (D · E) d1(D)∆ to D ◦ D , and

on C (C1,C2) and conjectured that it is non-degenerate. (Y · ∆) = d (D)(∆ · ∆) = d (D)(2 − 2g). 1 1 1 Note that

It remains to estimate (Y2 · ∆) where Y2 = D − Y1. Let 2 σ(D,D) = 2d1(D)d2(D) − (D ) = def(D). KC denote a positive canonical divisor on C, and let ϕ1,...,ϕg denote a basis for L(KC). For P ∈ C(k), let Castelnuovo (1906) proved the following theorem,

let D be a divisor on C ×C ; then σ(D,D) ≥ 0, with equality Φ(P) = det(ϕi(D j(P)) 1 2 if and only if D has valence zero.14 (Weil 1948a, II, n◦13, p. 52). Then Φ is a rational func- from which he was able to deduce Severi’s conjecture. tion on C, whose divisor we denote (Φ) = (Φ)0 − (Φ)∞. Of course, this all takes place in characteristic zero. The zeros of Φ correspond to the points (Di(P),D j(P)) As noted earlier, the Riemann-Roch theorem for with Di(P) = D j(P), i 6= j, and so surfaces in characteristic p was not available to Weil. The Italian proof of the complex Riemann-Roch theo- deg(Φ)0 = (Y2 · ∆). rem rests on a certain lemma of Enriques and Severi. Zariski (1952) extended this lemma to normal vari- On the other hand the poles of Φ are at the points P eties of all dimensions in all characteristics; in partic- for which D (P) lies in the support of K , and so j C ular, he obtained a proof of the Riemann-Roch theorem for normal surfaces in characteristic p. deg(Φ)∞ ≤ deg(KC)d1(D) = (2g − 2)d1(D). Mattuck and Tate (1958) used the Riemann-Roch Therefore, theorem in the case of a product of two curves to obtain a simple proof of the Castelnuovo-Severi inequal- 0 15 (D ◦ D ,∆) ≤ d1(D)(2 − 2g) + (2g − 2)d1(D) = 0 ity.

13 and so Serre writes (email July 2015): Weil always insisted that Artin’s conjecture on the holo- 0 def 0 0 0 σ(D◦D ) = d1(D◦D )+d2(D◦D )−(D◦D ,∆) ≥ 2gd1(D) ≥ 0. morphy of non-abelian L-functions is on the same level of difficulty as the Riemann hypothesis. In his book “Courbes Let D be a divisor on C×C. Then D becomes equiv- algébriques ...” he mentions (on p. 83) that the L-functions are polynomials, but he relies on the second volume for the alent to a divisor of the form considered in the last proof (based on the `-adic representations: the positivity re- paragraph after we pass to a finite generically Ga- sult alone is not enough). I find interesting that, while there lois covering V → C ×C (this follows from Weil 1948a, are several “elementary” proofs of the Riemann hypothesis for curves, none of them gives that Artin’s L-functions are Proposition 3, p. 43). Elements of the Galois group of polynomial. The only way to prove it is via `-adic cohomol- k(V) over k(C ×C) act on the matrix (ϕi(D j(P)) by per- ogy, or equivalently, `-adic representations. Curiously, the muting the columns, and so they leave its determi- situation is different over number fields, since we know sev- nant unchanged except possibly for a sign. On replac- eral non-trivial cases where Artin’s conjecture is true (thanks to Langlands theory), and no case where the Riemann hy- ing Φ with its square, we obtain a rational function on pothesis is! C×C, to which we can apply the above argument. This 14 completes the sketch. In fact, Castelnuovo proved a more precise result, which Kani (1984) extended to characteristic p, thereby obtaining Weil gives a rigorous presentation of the argu- another proof of the defect inequality in characteristic p. ment just sketched in II, pp. 42–54, of his book 1948a. 15 In their introduction, they refer to Weil’s 1940 note and In fact, he proves the stronger result: σ(ξ ◦ ξ 0) > 0 for write: “the [Castelnuovo-Severi] inequality is really a state-

20 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 In trying to understand the exact scope of Foundations of Algebraic Geometry (Weil 1946) the method of Mattuck and Tate, Grothendieck When Weil began the task of constructing foun- (1958a) stumbled on the Hodge index theorem.16 dations for his announcements he was, by his own In particular, he showed that the Castenuovo-Severi- Weil inequality follows from a general statement, account, not an expert in algebraic geometry. During valid for all surfaces, which itself is a simple con- a six-month stay in Rome, 1925–1926, he had learnt sequence of the Riemann-Roch theorem for sur- something of the Italian school of algebraic geometry, faces. but mainly during this period he had studied linear The proof presented in the preceding subsection functionals with Vito Volterra. incorporates these simplifications. In writing his Foundations, Weil’s main inspira- tion was the work of van der Waerden,17 which gives Variants a rigorous algebraic treatment of projective varieties Igusa (1949) gave another elaboration of the over fields of arbitrary characteristic and develops in- proof of the Riemann hypothesis for curves sketched tersection theory by global methods. However, Weil in Weil 1941. Following Castelnuovo, he first proves a was unable to construct the jacobian variety of a formula of Schubert, thereby giving the first rigorous curve as a projective variety. This forced him to in- proof of this formula valid over an arbitrary field. His troduce the notion of an abstract variety, defined by proof makes use of the general theory of intersection an atlas of charts, and to develop his intersection the- multiplicities in Weil’s Foundations. ory by local methods. Without the Zariski topology, Intersection multiplicities in which one of the fac- his approach was clumsy. However, his “abstract vari- tors is a hypersurface can be developed in an ele- eties” liberated algebraic geometry from the study of mentary fashion, using little more that the theory varieties embedding in an affine or projective space. of discrete valuations (cf. p. 17). In his 1953 thesis, In this respect, his work represents a break with the Weil’s student Frank Quigley “arranged” Weil’s 1941 past. proof so that it depends only on this elementary in- Weil completed his book in 1944. As Zariski wrote tersection theory (Quigley 1953). As did Igusa, he first in a review (BAMS 1948): proved Schubert’s formula. In the words of the author the main purpose of this book Finally, in his thesis (Hamburg 1951), Hasse’s stu- is “to present a detailed and connected treatment of the dent Roquette gave a proof of the Riemann hypoth- properties of intersection multiplicities, which is to include esis for curves based on Deuring’s theory of corre- all that is necessary and sufficient to legitimize the use made of these multiplicities in classical algebraic geometry, espe- spondences for double-fields (published as Roquette cially of the Italian school”. There can be no doubt whatso- 1953). ever that this purpose has been fully achieved by Weil. After a long and careful preparation (Chaps. I–IV) he develops in Applications to Exponential Sums two central chapters (V and VI) an intersection theory which for completeness and generality leaves little to be desired. It Davenport and Hasse (1935) showed that certain goes far beyond the previous treatments of this foundational arithmetic functions can be realized as the traces topic by Severi and van der Waerden and is presented with that absolute rigor to which we are becoming accustomed of Frobenius maps. Weil (1948c) went much further, in algebraic geometry. In harmony with its title the book is and showed that all exponential sums in one variable entirely self-contained and the subject matter is developed can be realized in this way. From his results on the ab initio. zeta functions of curves, he obtained new estimates It is a remarkable feature of the book that—with one ex- ception (Chap. III)—no use is made of the higher methods for these sums. Later developments, especially the of modern algebra. The author has made up his mind not to construction of `-adic and p-adic cohomologies, and assume or use modern algebra “beyond the simplest facts Deligne’s work on the zeta functions of varieties over about abstract fields and their extensions and the bare rudi- ments of the theory of ideals.” …The author justifies his profinite fields, have made this a fundamental tool in an- cedure by an argument of historical continuity, urging a re- alytic number theory. turn “to the palaces which are ours by birthright.” But it is very unlikely that our predecessors will recognize in Weil’s ment about the geometry on a very special type of surface— book their own familiar edifice, however improved and com- the product of two curves—and it is natural to ask whether it pleted. If the traditional geometer were invited to choose be- does not follow from the general theory of surfaces.” Appar- tween “makeshift constructions full of rings, ideals and val- ently they had forgotten that Weil had answered this question in 1941! 17 In the introduction, Weil writes that he “greatly profited 16 “En essayant de comprendre la portée exacte de leur from van der Waerden’s well-known series of papers (pub- méthode, je suis tombé sur l’énoncé suivant, connu en fait lished in Math. Ann. between 1927 and 1938)…; from Sev- depuis 1937 (comme me l’a signalé J. P. Serre), mais ap- eri’s sketchy but suggestive treatment of the same subject, paremment peu connu et utilisé.” The Hodge index theo- in his answer to van der Waerden’s criticism of the work of rem was first proved by analytic methods in Hodge 1937, the Italian school; and from the topological theory of inter- and by algebraic methods in Segre 1937 and in Bronowski sections, as developed by Lefschetz and other contemporary 1938. mathematicians.”

DECEMBER 2016 NOTICES OF THE ICCM 21 uations”18 on one hand, and constructions full of fields, lin- a simple natural proof of the Mordell-Weil theorem early disjoint fields, regular extensions, independent exten- for abelian varieties over global fields. sions, generic specializations, finite specializations and specializations of specializations on the other, he most proba- 1.13. Néron (1964) developed a theory of minimal bly would decline the choice and say: “A plague on both your models of abelian varieties over local and global houses!” fields, extending Kodaira’s theory for elliptic curves For fifteen years, Weil’s book provided a secure over function fields in one variable over C. foundation for work in algebraic geometry, but then 1.14. These developments made it possible to state was swept away by commutative algebra, sheaves, co- the conjecture of Birch and Swinnerton-Dyer for homology, and schemes, and was largely forgotten.19 abelian varieties over global fields (Tate 1966a). The For example, his intersection theory plays little role case of a jacobian variety over a global function field in Fulton 1984. It seems that the approach of van der inspired the conjecture of Artin-Tate concerning the Waerden and Weil stayed too close to the Italian origi- special values of the zeta function of a surface over a nal with its generic points, specializations, and so on; finite field (ibid. Conjecture C). what algebraic geometry needed was a complete ren- 1.15. Tate (1964) conjectured that, for abelian vari- ovation. eties A, B over a field k finitely generated over the prime field, the map Gal(ksep/k) Variétés Abéliennes et … (Weil 1948b) Z` ⊗ Hom(A,B) → Hom(T`A,T`B) In this book and later work, Weil constructs the is an isomorphism. Mumford explained to Tate that, for elliptic curves over a finite field, this follows from jacobian variety of a curve, and develops a com- the results of Deuring (1941). In one of his most beau- prehensive theory of abelian varieties over arbitrary tiful results, Tate proved the statement for all abelian fields, parallel to the transcendental theory over C. varieties over finite fields (Tate 1966b). At a key point Although inspired by his work on the Riemann hy- in the proof, he needed to divide a polarization by ln; pothesis, this work goes far beyond what is needed to for this he was able to appeal to “the proposition on justify his 1940 note. Weil’s book opened the door to the last page of Weil 1948b”. the arithmetic study of abelian varieties. In the twenty 1.16. (Weil-Tate-Honda theory) Fix a power q of a years following its publication, almost all of the im- prime p. An element π algebraic over Q is called a portant results on elliptic curves were generalized to Weil number if it is integral and |ρ(π)| = q1/2 for all em- abelian varieties. Before describing two of Weil’s most 0 beddings ρ : Q[π] → C. Two Weil numbers π and π are 0 important accomplishments in his book, I list some of conjugate if there exists an isomorphism Q[π] → Q[π ] these developments. sending π to π0. Weil attached a Weil number to each simple abelian variety over , whose conjugacy class 1.9. There were improvements to the theory of Fq depends only on the isogeny class of the variety, and abelian varieties by Weil and others; see (1.19) below. Tate showed that the map from isogeny classes of 1.10. Deuring’s theory of complex multiplication for simple abelian varieties over Fq to conjugacy classes elliptic curves was extended to abelian varieties of of Weil numbers is injective. Using the theory of com- arbitrary dimension by Shimura, Taniyama, and Weil plex multiplication, Honda (1968) shows that the map (see their talks at the Symposium on Algebraic Num- is also surjective. In this way, one obtains a classifica- ber Theory, Tokyo & Nikko, 1955). tion of the isogeny classes of simple abelian varieties over . Tate determined the endomorphism algebra 1.11. For a projective variety V, one obtains a height Fq of simple abelian variety A over in terms of its Weil function by choosing a projective embedding of V. In Fq number π: it is a central division algebra over [π] 1958 Néron conjectured that for an abelian variety Q which splits at no real prime of [π], splits at every there is a canonical height function characterized by Q finite prime not lying over p, and at a prime v above having a certain quadratic property. The existence of p has invariant such a height function was proved independently by Néron and Tate in the early 1960s. ordv(π) (9) inv (End0(A)) ≡ [ [π] : ](mod 1); v ord (q) Q v Qp 1.12. Tate studied the Galois cohomology of abelian v varieties, extending earlier results of Cassels for ellip- moreover, 0 1/2 tic curves. This made it possible, for example, to give 2dim(A) = [End (A): Q[π]] · [Q[π]: Q]. 0 18 Weil’s description of Zariski’s approach to the founda- Here End (A) = End(A) ⊗ Q. See Tate 1968. tions. 19 When Langlands decided to learn algebraic geometry in Iwasawa Theory Foretold Berkeley in 1964–1965, he read Weil’s Foundations… and Weil was very interested in the analogy between Conforto’s Abelsche Funktionen… A year later, as a student number fields and function fields and, in particular, at Harvard, I was able to attend Mumford’s course on algebraic geometry which used commutative algebra, sheaves, in “extending” results from function fields to number and schemes. fields. In 1942 he wrote:

22 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Our proof for the Riemann hypothesis depended upon a birational map f : V 99K G such that f (ab) = f (a) f (b) the extension of the function-fields by roots of unity, i.e., whenever ab is defined; the pair (G, f ) is unique up to a by constants; the way in which the Galois group of such ex- ◦ tensions operates on the classes of divisors in the original unique isomorphism. (Weil 1948b, n 33, Thm 15; Weil field and its extensions gives a linear operator, the charac- 1955.) teristic roots (i.e., the eigenvalues) of which are the roots of (g) the zeta-function. On a number field, the nearest we can get Let C denote the symmetric product of g copies to this is by the adjunction of ln-th roots of unity, l being of C, i.e., the quotient of Cg by the action of the sym- fixed; the Galois group of this infinite extension is cyclic, metric group Sg. It is a smooth variety of dimension and defines a linear operator on the projective limit of the (g) (absolute) class groups of those successive finite extensions; g over k. The set C (k) consists of the unordered this should have something to do with the roots of the zeta- g-tuples of closed points on C, which we can regard function of the field. (Letter to Artin, Œuvres I, p. 298, 1942.) as positive divisors of degree g on C. Fix a P ∈ C(k). Let D be a positive divisor of degree g on C. According to the Riemann-Roch theorem Construction of the Jacobian Variety of a Curve Let C be a nonsingular projective curve over a l(D) = 1 + l(KC − D) ≥ 1, field k, which for simplicity we take to be algebraically and one can show that equality holds on a dense open closed. The jacobian variety J of C should be such that subset of C(g). Similarly, if D is a positive divisor of de- J(k) is the group Jac(C) of linear equivalence classes of gree 2g on C, then l(D−gP) ≥ 1 and equality holds on a divisors on C of degree zero. Thus, the problem Weil nonempty open subset U0 of C(2g). Let U be the inverse faced was that of realizing the abstract group Jac(C) image of U0 under the obvious map C(g) ×C(g) → C(2g). as a projective variety in some natural way—over C Then U is a dense open subset of C(g) ×C(g) with the the theta functions provide a projective embedding property that l(D + D0 − gP) = 1 for all (D,D0) ∈ U(k). of Jac(C). He was not able to do this, but as he writes Now let (D,D0) ∈ U(k). Because l(D + D0 − gP) > 0, (Œuvres I, p. 556): there exists a positive divisor D00 on C linearly equiv- 0 0 In the spring of 1941, I was living in Princeton… I often alent to D + D − gP, and because l(D + D − gP) = 1, the 00 worked in Chevalley’s office in Fine Hall; of course he was divisor D is unique. Therefore, there is a well-defined aware of my attempts to “define” the jacobian, i.e., to con- law of composition struct algebraically a projective embedding. One day, com- ing into his office, I surprised him by telling him that there (D,D0) 7→ D00 : U ×U → C(g)(k). was no need; for the jacobian everything comes down to its local properties, and a piece of the jacobian, joined to the Theorem 1.18. There exists a unique rational map group property (the addition of divisor classes) suffices am- ply for that. The idea came to me on the way to Fine Hall. (g) (g) (g) It was both the concept of an “abstract variety” which had m: C ×C 99K C just taken shape, and the construction of the jacobian as an algebraic group.20 whose domain of definition contains the subset U and which is such that, for all fields K containing k and all In order to explain Weil’s idea, we need the notion (D,D0) in U(K), m(D,D0) ∼ D+D0 −gP; moreover m makes of a birational group over k. This is a nonsingular vari- C(g) into a birational group. ety V together with a rational map m: V ×V 99K V such that This can be proved, according to taste, by using generic points (Weil 1948b) or functors (Milne 1986). (a) m is associative (that is, (ab)c = a(bc) whenever On combining the two theorems, we obtain a group both terms are defined); variety J over k birationally equivalent to C(g) with its (b) the rational maps (a,b) 7→ (a,ab) and (a,b) 7→ (b,ab) partial group structure. This is the jacobian variety. from V ×V to V ×V are both birational. Notes Theorem 1.17. Let (V,m) be a birational group V 1.19. Weil (1948b Thm 16, et seqq.) proved that the over k. Then there exists a group variety G over k and jacobian variety is complete, a notion that he him-

20 self had introduced. Chow (1954) gave a direct con- En ce printemps de 1941, je vivais à Princeton… Je tra- struction of the jacobian variety as a projective va- vaillais souvent dans le bureau de Chevalley à Fine Hall; bien entendu il était au courant de mes tentatives pour “définir” riety over the same base field as the curve. Weil la jacobienne, c’est-à-dire pour en construire algébriquement (1950) announced the existence of two abelian vari- un plongement projectif. Un jour, entrant chez lui, je le sur- eties attached to a complete normal algebraic vari- pris en lui disant qu’il n’en était nul besoin; sur la jacobi- ety, namely, a Picard variety and an Albanese variety. enne tout se ramène à des propriétés locales, et un morceau For a curve, both varieties equal the jacobian variety, de jacobienne, joint à la propriété de groupe (l’addition des but in general they are distinct dual abelian varieties. classes de diviseurs) y suffit amplement. L’idée m’en était This led to a series of papers on these topics (Mat- venue sur le chemin de Fine Hall. C’était à la fois la notion de “variété abstraite” qui venait de prendre forme, et la con- susaka, Chow, Chevalley, Nishi, Cartier, …) culminat- struction de la jacobienne en tant que groupe algébrique, ing in Grothendieck’s general construction of the Pi- telle qu’elle figure dans [1948b]. card scheme (see Kleiman 2005).

DECEMBER 2016 NOTICES OF THE ICCM 23 1.20. Weil’s construction of an algebraic group from The Characteristic Polynomial of an Endomorphism a birational group has proved useful in other con- Proposition 1.21. For all integers n ≥ 1, the map texts, for example, in the construction of the Néron 2g model of abelian variety (Néron 1964; Artin 1964, nA : A → A has degree n . Therefore, for ` 6= char(k), the def al -module T A = lim A n (k ) is free of rank 2g. 1986; Bosch et al. 1990, Chapter 5). Z` ` ←−n ` Proof. Let D be an ample divisor on A (e.g., a hy- The Endomorphism Algebra of an Abelian Variety perplane section under some projective embedding); then Dg is a positive zero-cycle, and (Dg) 6= 0. After The exposition in this subsection includes im- ∗ possibly replacing D with D + (−1)AD, we may sup- provements explained by Weil in his course at the ∗ pose that D is symmetric, i.e., D ∼ (−1)AD. An induc- University of Chicago, 1954–1955, and other articles, ∗ 2 tion argument using (10) shows that nAD ∼ n D, and which were incorporated in Lang 1959. In particular, so we use that an abelian variety admits a projective em- ∗ g ∗ ∗ 2 2 2g g bedding, and we use the following consequence of the (nAD ) = (nAD · ... · nAD) ∼ (n D · ... · n D) ∼ n (D ). theorem of the cube: Let f ,g,h be regular maps from a ∗ 2g variety V to an abelian variety A, and let D be a divisor Therefore nA has degree n . 2 on A; then A map f : W → Q on a vector space W over a field Q is said to be polynomial (of degree d) if, for every (10) ( f + g + h)∗D − ( f + g)∗D − (g + h)∗D − ( f + h)∗D finite linearly independent set {e1,...,en} of elements + f ∗D + g∗D + h∗D ∼ 0. of V,

We shall also need to use the intersection theory for f (a1e1 + ··· + anen) = P(a1,...,an), xi ∈ Q, divisors on smooth projective varieties. As noted earlier, this is quite elementary. for some P ∈ Q[X1,...,Xn] (of degree d). To show that a map f is polynomial, it suffices to check that, for all Review of the Theory Over C v,w ∈ W, the map x 7→ f (xv + w): Q → Q is a polynomial in x. Let A be an abelian variety of dimension g over C. Then A(C) ' T/Λ where T is the tangent space to A Lemma 1.22. Let A be an abelian variety of dimen- 0 at 0, and Λ = H1(A,Z). The endomorphism ring End(A) sion g. The map α 7→ deg(α): End (A) → Q is a polyno- of A acts faithfully on Λ, and so it is a free Z-module mial function of degree 2g on End0(A). of rank ≤ 4g. We define the characteristic polynomial 2g Proof. Note that deg(nβ) = deg(nA)deg(β) = n deg(β), Pα (T) of an endomorphism α of A to be its character- and so it suffices to prove that deg(nβ + α) (for n ∈ Z istic polynomial det(T − α | Λ) on Λ. It is the unique and β,α ∈ End(A)) is polynomial in n of degree ≤ 2g. polynomial in [T] such that Z Let D be a symmetric ample divisor on A. A direct calculation using (10) shows shows that Pα (m) = deg(m − α) (11) deg(nβ +α)(Dg) for all m ∈ . It can also be described as the character- Z g g 21 = (n(n − 1)) (D ) + terms of lower degree in n. istic polynomial of α acting on A(C)tors ' (Q ⊗ Λ)/Λ. Choose a Riemann form for A. Let H be the associ- As (Dg) 6= 0 this completes the proof. 2 ated positive-definite hermitian form on the complex vector space T and let α 7→ α† be the associated Rosati Theorem 1.23. Let α ∈ End(A). There is a unique 0 def involution on End (A) = End(A) ⊗ Q. Then monic polynomial Pα (T) ∈ Z[T] of degree 2g such that Pα (n) = deg(nA − α) for all integers n. H(αx,y) = H(x,α†y), all x,y ∈ T, α ∈ End0(A), Proof. If P1 and P2 both have this property, then P1 − P has infinitely many roots, and so is zero. For the and so † is a positive involution on End0(A), i.e., the 2 existence, take = 1 in (11). 2 trace pairing β A

We call Pα the characteristic polynomial of α and † 0 0 (α,β) 7→ Tr(α ◦ β ): End (A) × End (A) → Q we define the trace Tr(α) of α by the equation

2g 2g−1 is positive definite (see, for example, Rosen 1986). Pα (T) = T − Tr(α)T + ··· + deg(α). Remarkably, Weil was able to extend these statements to abelian varieties over arbitrary base fields. The Endomorphism Ring of an Abelian Variety

21 g Let A and B be abelian varieties over k, and let ` be Choose an isomorphism (Q⊗Λ/Λ) → (Q/Z)2 , and note that End(Q/Z) ' Zˆ . The action of α on Q⊗Λ/Λ defines an element a prime 6= char(k). The family of `-power torsion points ˆ al of M2g(Z), whose characteristic polynomial is Pα (T). in A(k ) is dense, and so the map

24 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Hom(A,B) → HomZ` (T`A,T`B) Proof. For an endomorphism α of A, is injective. Unfortunately, this doesn’t show that (14) |det(T`α)|` = |deg(α)|` . Hom(A,B) is finitely generated over Z. To continue, we shall need an elementary lemma (Weil n Lemma 1.24. Let α ∈ Hom(A,B); if α is divisible by ` in 1948b, n◦68, Lemme 12): Hom(T A,T B), then it is divisible by `n in Hom(A,B). ` ` al A polynomial P(T) ∈ Q`[T], with roots a1,...,ad in Q` , is d Proof. For each n, there is an exact sequence uniquely determined by the numbers ∏i=1 F(ai) ` as F runs through the elements of Z[T]. `n 0 → A n → A −→ A → 0. ` For the polynomial Pα (T),

The hypothesis implies that α is zero on A`n , and so d `n it factors through the quotient map A −→ A. 2 ∏F(ai) = ±deg(F(α)), i=1 Theorem 1.25. The natural map and for the characteristic polynomial Pα,`(T) of α on T`A, (12) Z` ⊗ Hom(A,B)→Hom(T`A,T`B)

d is injective (with torsion-free cokernel). Hence ∏F(ai) = |det(F(T`α))|` . i=1 Hom(A,B) is a free Z-module of finite rank ≤ 4dim(A) ` dim(B). Therefore, (14) shows that Pα (T) and Pα,`(T) coincide as elements of T . 2 Proof. The essential case is that with A = B and A sim- Q`[ ] ple. Let e1,...,em be elements of End(A) linearly inde- Positivity pendent over ; we have to show that T (e ),...,T (e ) Z ` 1 ` m An ample divisor D on A defines an isogeny are linearly independent over . def Z` ϕ : A → A∨ from A to the dual abelian variety A∨ = Let M (resp. M) denote the -module (resp. D Q Z Pic0(A), namely, a 7→ [D − D] where D is the translate -vector space) generated in End0(A) by the e . Be- a a Q i of D by a. A polarization of A is an isogeny λ : A → A∨ cause A is simple, every nonzero endomorphism α that becomes of the form ϕ over kal. As λ is an of A is an isogeny, and so deg(α) is an integer > 0. D isogeny, it has an inverse in Hom0(A∨,A). The Rosati The map deg: M → is continuous for the real topol- Q Q involution on End0(A) corresponding to λ is ogy because it is a polynomial function (1.22), and so def U = {α ∈ QM | deg(α) < 1} is an open neighbourhood α 7→ α† def= λ −1 ◦ α∨ ◦ λ. of 0. As It has the following properties: (QM ∩ End(A)) ∩U ⊂ End(A) ∩U = 0, † † † † † † † (α + β) = α + β , (αβ) = β α , a = a for all a ∈ Q. we see that QM ∩ End(A) is discrete in QM, which im- def plies that it is finitely generated as a Z-module. Hence Theorem 1.27. The Rosati involution on E = End0(A) there exists an integer N > 0 such that is positive, i.e., the pairing

† (13) N(QM ∩ End(A)) ⊂ M. α,β 7→ TrE/Q(α ◦ β ): E × E → Q

Suppose that there exist ai ∈ Z`, not all zero, such is positive definite. that a T (e ) = 0. For a fixed m ∈ , we can find inte- ∑ i ` i N Proof. We have to show that Tr(α ◦α†) > 0 for all α 6= 0. gers ni sufficiently close to the ai that the sum ∑niei is Let D be the ample divisor corresponding to the polar- divisible by `m in (T A), and hence in (A). There- End ` End ization used in the definition of †. A direct calculation fore shows that m ∑(ni/` )ei ∈ QM ∩ End(A), 2g Tr(α ◦ α†) = (Dg−1 · α−1(D)) m m (Dg) and so N ∑(ni/` )ei ∈ M, i.e., niN/` ∈ Z and ord`(ni) + ord (N) ≥ m for all i. But if n is close to a , then ord (n ) = ` i i ` i (Lang 1959, V, §3, Thm 1). I claim that (Dg−1 ·α−1(D)) > ord (a ), and so ord (a ) + ord (N) ≥ m. As m was arbi- ` i ` i ` 0. We may suppose that D is a hyperplane section of trary, we have a contradiction. 2 A relative to some projective embedding. There ex- The `-Adic Representation ist hyperplane sections H1,...,Hg−1 such that H1 ∩... ∩ H ∩α−1D has dimension zero. By dimension theory, Proposition 1.26. For all ` 6= char(k),P (T) is the char- g−1 α the intersection is nonempty, and so acteristic polynomial of α acting on T`A; in particular, −1 det(α | T`A) = deg(α). (H1 · ... · Hg−1 · α D) > 0.

DECEMBER 2016 NOTICES OF THE ICCM 25 g−1 −1 As D ∼ Hi for all i, we have (D ·α D) = (H1 ·...·Hg−1 · The Riemann hypothesis for a curve follows from α−1D) > 0. 2 applying (1.30) to the jacobian variety of C. Recall that the radical rad(R) of a ring R is the in- Summary 1.31. Let A be an abelian variety of dimen- tersection of the maximal left ideals in R. It is a two- sion g over k = Fq, and let P(T) be the characteristic sided ideal, and it is nilpotent if R is artinian. An alge- polynomial of the Frobenius endomorphism π. Then bra R of finite dimension over a field Q is semisimple P(T) is a monic polynomial of degree 2g with inte- if and only if (R) = . rad 0 ger coefficients, and its roots a1,...,a2g have absolute 0 1/2 Corollary 1.28. The Q-algebra End (A) is semisimple. value q . For all ` 6= p, Proof. If not, the radical of End0(A) contains a nonzero def P(T) = det(T − π | VÀ), VÀ = TÀ ⊗ Q`. element α. As β def= αα† has nonzero trace, it is a nonzero element of the radical. It is symmetric, and For all n ≥ 1, so β 2 = ββ † 6= 0, β 4 = (β 2)2 6= 0, … . Therefore β is not 2g nilpotent, which is a contradiction. 2 n (15) ∏(1 − ai ) = |A(kn)| where [kn : k] = n, Corollary 1.29. Let α be an endomorphism of A such i=1 † that α ◦ α = qA, q ∈ Z. For every homomorphism and this condition determines P. We have ρ : Q[α] → C, P(T) = qg · T 2g · P(1/qT). ρ(α†) = ρ(α) and |ρα| = q1/2. Notes. Essentially everything in this subsection is Proof. As [α] is stable under †, it is semisimple, and Q in Weil 1948b, but, as noted at the start, some of hence a product of fields. Therefore R⊗Q[α] is a prod- † the proofs have been simplified by using Weil’s later uct of copies of R and C. The involution on Q[α] ex- work. In 1948 Weil didn’t know that abelian varieties tends by continuity to an involution of R ⊗ Q[α] hav- are projective, and he proved (1.27) first for jaco- † ing the property that Tr(β ◦ β ) ≥ 0 for all β ∈ R ⊗ Q[α] bians, where the Rosati involution is obvious. The with inequality holding on a dense subset. The only more direct proof of (1.27) given above is from Lang such involution preserves the each factor and acts as 1957. the identity on the real factors and as complex conjugation on the complex factors. This proves the first The Weil Conjectures (Weil 1949) statement, and the second follows:

† 2 Weil studied equations of the form q = ρ(qA) = ρ(α ◦ α ) = ρ(α) · ρ(α) = ρ(α) . 2

m0 mr (16) a0X0 + ··· + arXr = b Theorem 1.30. Let A be an abelian variety over k = Fq, and let π ∈ End(A) be the Frobenius endomorphism. Let over a finite field k, and obtained an expression in † be a Rosati involution on End0(A). For every homo- terms of Gauss sums for the number of solutions in k. morphism ρ : Q[π] → C, Using a relation, due to Davenport and Hasse (1935), between Gauss sums in a finite field and in its exten- ρ(α†) = ρ(π) and |ρπ| = q1/2. sions, he was able to obtain a simple expression for Proof. This will follow from (1.29) once we show that the formal power series (“generating function”) † π ◦π = qA. This can be proved by a direct calculation, ∞ n but it is more instructive to use the Weil pairing. Let ∑(# solutions of (16) in kn)T . λ be the polarization defining †. The Weil pairing is a 1 nondegenerate skew-symmetric pairing In the homogeneous case λ e : TlA × TlA → TlGm m m (17) a0X0 + ··· + arXr = 0, with the property that he found that λ λ † e (αx,y) = e (x,α y)(x,y ∈ TlA, α ∈ End(A)). ∞ n−1 d 1 ∑NnT = log r−1 Now 1 dT (1 − T)···(1 − q T) d eλ (x,(π† ◦ π)y) + (−1)r logP(T) dT λ λ λ λ = e (πx,πy) = πe (x,y) = qe (x,y) = e (x,qAy). where P(T) is a polynomial of degree A equal to the † Therefore π ◦ π and qA agree as endomorphisms of number of solutions in rational numbers αi of the sys- TlA, and hence as endomorphisms of A. 2 tem nαi ≡ 1, ∑αi ≡ 0 (mod 1), 0 < αi < 1. Dolbeault was

26 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 able to tell Weil that the Betti numbers of a hypersur- a curve C, the Betti numbers are 1, 2g, 1 where g is face defined by an equation of the form (17) over C the smallest natural number for which the Riemann are the coefficients of the polynomial inequality

1 + T 2 + ··· + T 2r−2 + AT r. l(D) ≥ deg(D) − g + 1 holds, but for dimension > 1? As he wrote (1949, p. 409): No cohomology groups appear in Weil’s article This, and other examples which we cannot discuss here, and no cohomology theory is conjectured to exist,23 seems to lend some support to the following conjectural but Weil was certainly aware that cohomology pro- statements, which are known to be true for curves, but which vides a heuristic explanation of (W1–W3). For exam- I have not so far been able to prove for varieties of higher dimension.22 ple, let ϕ be a finite map of degree δ from a nonsingular complete variety V to itself over C. For each n, let Weil Conjectures. Let V be a nonsingular projective Nn be the number of solutions, supposed finite, of the variety of dimension d defined over a finite field k equation P = ϕn(P) or, better, the intersection num- with q elements. Let Nn denote the number points of ber (∆ · Γϕn ). Then standard arguments show that the V rational over the extension of k of degree n. Define power series Z(T) defined by (18) satisfies the func- the zeta function of C to be the power series Z(V,T) ∈ tional equation Q[[T]] such that 1/2 χ ∞ (21) Z(1/δT) = ±(δ T) · Z(T) d logZ(V,T) n−1 (18) = ∑NnT . dT 1 where χ is the Euler-Poincaré characteristic of V (Weil, Œuvres I, p. 568). For a variety V of dimension d, the Then: Frobenius map has degree qd, and so (21) suggests (W1) (rationality) Z(V,T) is a rational function in T; (19). (W2) (functional equation) Z(V,T) satisfies the func- Example 1.32. Let A be an abelian variety of dimen- tional equation sion g over a field k with q elements, and let a1,...,a2g (19) Z(V,1/qdT) = ±qdχ/2 · T χ · Z(V,T) be the roots of the characteristic polynomial of the Frobenius map. Let Pr(T) = ∏(1 − ai,rT) where the ai,r with χ equal to the Euler-Poincaré characteristic run through the products of V (intersection number of the diagonal with a ···a , 0 < i < ··· < i ≤ 2g. itself); i1 ir 1 r (W3) (integrality) we have It follows from (15) that

P1(T)···P2d−1(T) P1(T)···P2g−1(T) (20) Z(V,T) = d (1 − T)P3(T)···(1 − q T) Z(A,T) = g . (1 − T)P3(T)···(1 − q T)

with Pr(T) ∈ Z[T] for all r; The rth Betti number of an abelian variety of dimen- Br (W4) (Riemann hypothesis) write Pr(T) = ∏1 (1−αriT); g sion g over C is (r), which equals deg(Pr), and so the then Weil conjectures hold for A. r/2 |αri| = q Aside 1.33. To say that Z(V,T) is rational means that for all r, i; there exist elements c1,...,cm ∈ Q, not all zero, and an (W5) (Betti numbers) call the degrees Br of the poly- integer n0 such that, for all n ≥ n0, nomials Pr the Betti numbers of V; if V is obtained by reduction modulo a prime ideal p in c1Nn + c2Nn+1 + ··· + cmNn+m−1 = 0. a number field K from a nonsingular projective In particular, the set of Nn can be computed induc- variety V˜ over K, then the Betti numbers of V tively from a finite subset. are equal to the Betti numbers of V˜ (i.e., of the complex manifold V˜ (C)). 2. Weil Cohomology Weil’s considerations led him to the conclusion, startling at the time, that the “Betti numbers” of an After Weil stated his conjectures, the conven- algebraic variety have a purely algebraic meaning. For tional wisdom eventually became that, in order to

22 Weil’s offhand announcement of the conjectures misled 23 Weil may have been aware that there cannot exist a good one reviewer into stating that “the purpose of this paper is to cohomology theory with Q-coefficients; see (2.2). Recall also give an exposition of known results concerning the equation that Weil was careful to distinguish conjecture from specu- m0 mr a0X0 + ··· + arXr = b” (MR 29393). lation.

DECEMBER 2016 NOTICES OF THE ICCM 27 prove the rationality of the zeta functions, it was Künneth formula. Let p,q: V ×W ⇒ V,W be the pro- necessary to define a good “Weil cohomology the- jection maps. Then the map ory” for algebraic varieties. In 1959, Dwork surprised ∗ ∗ ∗ ∗ ∗ everyone by finding an elementary proof, depend- x ⊗ y 7→ p x · q y : H (V) ⊗ H (W) → H (V ×W) ing only on p-adic analysis, of the rationality of the is an isomorphism of graded Q-algebras. zeta function of an arbitrary variety over a finite field (Dwork 1960).24 Nevertheless, a complete under- Cycle map. There are given group homomorphisms standing of the zeta function requires a cohomology cl : Cr (V) → H2r(V)(r) theory, whose interest anyway transcends zeta func- V rat tions, and so the search continued. satisfying the following conditions: After listing the axioms for a Weil cohomology (a) (functoriality) for every regular map φ : V → W, theory, we explain how the existence of such a theory implies the first three Weil conjectures. Then we ∗ ∗ φ ◦ clW = clV ◦ φ describe the standard Weil cohomologies. (23) φ∗ ◦ clV = clW ◦ φ∗ r For a smooth projective variety V, we write Crat(V) r for the Q-vector space of algebraic cycles of codimen- (b) (multiplicativity) for every Y ∈ Crat(V) and Z ∈ s sion r (with Q-coefficients) modulo rational equiva- Crat(W), lence. clV×W (Y × Z) = clV (Y) ⊗ clW (Z).

∗ The Axioms for a Weil Cohomology Theory (c) (normalization) If P is a point, so that Crat(P) ' Q ∗ and H (P) ' Q, then clP is the natural inclusion We fix an algebraically closed “base” field k and a map. “coefficient” field Q of characteristic zero. A Weil co- ∗ Let φ : V → W be a regular map of nonsingular homology theory is a contravariant functor V H (V) projective varieties over k, and let ∗ be the map from the category of nonsingular projective varieties φ H∗(φ): H∗(W) → H∗(V). Because the pairing (22) is non- over k to the category of finite-dimensional, graded, degenerate, there is a unique linear map anti-commutative Q-algebras carrying disjoint unions to direct sums and admitting a Poincaré duality, a ∗ ∗−2c φ∗ : H (V) → H (W)(−c), c = dimV − dimW Künneth formula, and a cycle map in the following sense. such that the projection formula

∗ Poincaré duality. Let V be connected of dimen- ηW (φ∗(x) · y) = ηV (x · φ y) sion d. holds for all x ∈ Hr(V), y ∈ H2 dimV−r(W)(dimV). This r (a) The Q-vector spaces H (V) are zero except for 0 ≤ explains the maps on the left of the equals signs 2d r ≤ 2d, and H (V) has dimension 1. in (23), and the maps on the right refer to the 2 1 (b) Let Q(−1) = H (P ), and let Q(1) denote its dual. standard operations on the groups of algebraic cy- For a Q-vector space V and integer m, we let V(m) cles modulo rational equivalence (Fulton 1984, Chap- ⊗m ⊗−m equal V ⊗Q Q(1) or V ⊗Q Q(−1) according as ter I). m is positive or negative. For each V, there is given 2d The Lefschetz Trace Formula a natural isomorphism ηV : H (V)(d) → Q. (c) The pairings Let H∗ be a Weil cohomology over the base field k, and let V be nonsingular projective variety over k. We r 2d−r 2d (22) H (V) × H (V)(d) → H (V)(d) ' Q use the Künneth formula to identify H∗(V × V) with H∗(V) ⊗ H∗(V). ∗ induced by the product structure on H (V) are Let φ : V → V be a regular map. We shall need an non-degenerate. expression for the class of the graph Γφ of φ in

24 “His method consists in assuming that the variety is a hy- 2d persurface in affine space (every variety is birationally iso- H2d(V ×V)(d) ' MHr(V) ⊗ H2d−r(V)(d). morphic to such a hypersurface, and an easy unscrewing r=0 lets us pass from there to the general case); in this case he does a computation with “Gauss sums” analogous to that r r 2d−r Let (ei )i be a basis for H (V) and let ( fi )i be the of Weil for an equation a xni = b. Of course, Weil himself ∑ i i dual basis in H2d−r(V)(d); we choose e0 = 1, so that had tried to extend his method and had got nowhere…” 2d Serre, Grothendieck-Serre Correspondence p. 102. Dwork ex- ηV ( f ) = 1. Then pressed Z(V,T) as a quotient of two p-adically entire func- ∗ r 2d−r tions, and showed that every such function with nonzero ra- clV×V (Γφ ) = ∑φ (ei ) ⊗ fi . dius of convergence is rational. r,i

28 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Theorem 2.1 (Lefschetz trace formula). Let φ : V → V0/k0). We assume that there exists a Weil cohomology V be a regular map such that Γφ · ∆ is defined. Then theory over k with coefficients in Q.

2d Proposition 2.3. The zeta function r r (Γφ · ∆) = (−1) Tr(φ | H (V)) ∑ P (T)···P (T) r=0 (24) Z(V,T) = 1 2d−1 (1 − T)P (T)···(1 − qdT) Proof. We the above notations, we have 2 with P (V,T) = det(1 − T | Hr(V,Q)). ∗ r 2d−r r π clV×V (Γφ ) = ∑φ (ei ) ⊗ fi and r,i Proof. Recall that r d−r cl (∆) = e ⊗ f 2 n V×V ∑ i i def T r,i logZ(V,T) = N . ∑ n n r(2d−r) 2d−r r n>0 = ∑(−1) fi ⊗ ei r,i The fixed points of πn have multiplicity 1, and so = (−1)r f 2d−r ⊗ er. ∑ i i 2d r,i (2.1) r n r Nn = (Γπn · ∆) = ∑(−1) Tr(π | H (V,Q)). Thus r=0

r ∗ r 2d−r 2d The following elementary statement completes the clV×V (Γφ · ∆) = ∑(−1) φ (ei ) fi ⊗ f r,i proof: Let α be an endomorphism of a finite-dimen- 2d sional vector space V; then = (− )r ( ∗)( f 2d ⊗ f 2d) ∑ 1 Tr φ T n r=0 (25) log(det(1 − αT | V)) = − ∑ Tr(αn | V) . 2 n≥1 n ∗ r 2d−r r ∗ r because φ (ei ) f j is the coefficient of e j when φ (ei ) is expressed in terms of the basis (er ). On applying j Proposition 2.4. The zeta function Z(V,T) ∈ Q[T]. ηV×V to both sides, we obtain the required formula. 2 Proof. We know that Z(V,T) ∈ Q[T]. To proceed, we This is Lefschetz’s original 1924 proof (see Steen- shall need the following elementary criterion: rod 1957, p. 27). i Let f (T) = ∑i≥0 aiT ∈ Q[[T]]; then f (T) ∈ Q[T] if and only if There Is no Weil Cohomology Theory with Coeﬃcients in there exist integers m and n0 such that the Hankel determinant Qp or R. an an+1 ··· an+m−1

2.2. Recall (1.16) that for a simple abelian variety A an+1 an+2 ··· an+m over a finite field k, E def= End0(A) is a division algebra an+2 an+3 ··· an+m+1 . . . def 1/2 . . . with centre F = Q[π], and 2dim(A) = [E : F] ·[F : Q]. Let Q be a field of characteristic zero. In order for E to act an+m−1 an+m ··· an+2m−2 on a Q-vector space of dimension 2dim(A), the field Q is 0 for all n ≥ n0 (this is a restatement of (1.33); it is an exercise in Chap. 4 of Bourbaki’s Algèbre). must split E, i.e., Q⊗Q E must be a matrix algebra over Q ⊗Q F. The endomorphism algebra of a supersingu- The power series Z(V,T) satisfies this criterion in lar elliptic curve over a finite field containing 2 is a Fp Q[[T]], and hence also in Q[[T]]. 2 division quaternion algebra over Q that is nonsplit exactly at p and the real prime (Hasse 1936). Therefore, As in the case of curves, the zeta function can be there cannot be a Weil cohomology with coefficients written in or (hence not in either).25 Tate’s formula 1 Qp R Q Z(V,T) = (9) doesn’t forbid there being a Weil cohomology with ∏ deg(v) v∈|V| 1 − T coeffients in Q`, ` 6= p, or in the field of fractions of the Witt vectors over k. That Weil cohomology theo- where v runs over the set |V| of closed points of V (as ries exist over these fields (see below) is an example a scheme). Hence of Yhprum’s law in mathematics: everything that can 2 work, will work. Z(V,T) = 1 + a1T + a2T + ···

with the ai ∈ Z. When we write Proof of the Weil Conjectures W1–W3 P(T) Let V be a nonsingular projective variety of di- Z(V,T) = , P,R ∈ [T], gcd(P,R) = 1, 0 R(T) Q mension d over k0 = Fq, and let V be the variety obtained by extension of scalars to the algebraic closure we can normalize P and R so that k of k0. Let π : V → V be the Frobenius map (relative to P(T) = 1 + b1T + ··· ∈ Q[T] 25 It was Serre who explained this to Grothendieck, sometime in the 1950s. R(T) = 1 + c1T + ··· ∈ Q[T].

DECEMBER 2016 NOTICES OF THE ICCM 29 Proposition 2.5. Let P,R be as above. Then P and R are formal sum Z = ∑niZi with ni ∈ Z and Zi an irreducible uniquely determined by V, and they have coefficients closed subvariety of codimension r; such cycles form r in Z. a group C (V). Proof. The uniqueness follows from unique factor- 2.9. Let ∼ be an adequate equivalence relation on r ∗ def L r ization in Q[T]. If some coefficient of R is not an in- the family of groups C (V). Then C∼(V) = r≥0 C / ∼ teger, then β −1 is not an algebraic integer for some becomes a graded ring under intersection product; moreover, push-forwards and pull-backs of algebraic root β of R. Hence ordl(β) > 0 for some prime l, and cycles with respect to regular maps are well-defined. Z(V,β) = 1 + a β + a β 2 + ··· converges l-adically. This 1 2 Let Ar (V) = Cr (V) ⊗ . There is a bilinear map contradicts the fact that β is a pole of Z(V,T). We have ∼ ∼ Q −1 dim(V1)+r dim(V2)+s dim(V1)+r+s shown that R(T) has coefficients in Z. As Z(V,T) ∈ A∼ (V1×V2)×A∼ (V2×V3) → A∼ (V1×V3) Z[[T]], the same argument applies to P(T). 2 sending ( f ,g) to Proposition 2.6. We have def ∗ ∗ g ◦ f = (p13)∗(p12 f · p23g) Z(V,1/qdT) = ±qdχ/2 · T χ · Z(V,T) where pi j is the projection V1 ×V2 ×V3 → Vi ×Vj. This where χ = (∆ · ∆). is associative in an obvious sense. In particular, dim(V) A∼ (V ×V) is a Q-algebra. Proof. The Frobenius map π has degree qd. Therefore, for any closed point P of V, π∗P = qdP, and 2.10. Two algebraic r-cycles f ,g are numerically equivalent if ( f · h) = (g · h) for all (d − r)-cycles h for ∗ ∗ d d which the intersection products are defined. This is (26) π clV (P) = clV (π P) = clV (q P) = q clV (P). an adequate equivalence relation, and so we get a d 2d d Thus π acts as multiplication by q on H (V)(d). Q-algebra Anum(V ×V). From this, and Poincaré duality, it follows that if Let H be a Weil cohomology theory with coeffi- r α1,...,αs are the eigenvalues of π acting on H (V), cient field Q, and let Ar (V) (resp. Ar (V,Q)) denote the d d H H then q /α1,...,q /αs are the eigenvalues of π acting on -subspace (resp. Q-subspace) of H2r(V)(r) spanned 2d−r Q H (V). An easy calculation now shows that the re- by the algebraic classes. quired formula holds with χ replaced by r 2.11. The Q-vector space Anum(V) is finite-dimen- r r d−r (−1) dimH (V), sional. To see this, let f1,..., fs be elements of AH (V) ∑ d−r 2d−2r spanning the Q-subspace AH (V,Q) of H (V)(d −r); but the Lefschetz trace formula (2.1) with φ = id shows then the kernel of the map that this sum equals (∆ · ∆). 2 r s x 7→ (x · f1,...,x · fs): AH (V) → Q 2.7. In the expression (24) for Z(V,T), we have not consists of the elements of Ar (V) numerically equiv- shown that the polynomials Pr have coefficients in H alent to zero, and so its image is Ar (V). Q nor that they are independent of the Weil coho- num mology theory. However, (2.5) shows that this is true r r 2.12. Let Anum(V,Q) denote the quotient of AH (V,Q) by for the numerator and denominator after we have re- the left kernel of the pairing moved any common factors. If the Pr(T) are relatively r d−r d prime in pairs, then there can be no cancellation, and AH (V,Q) × AH (V,Q) → AH (V,Q) ' Q. i Pr(T) = 1 + ∑ ar,iT has coefficients in and is inde- i Z Then Ar (V) → Ar (V,Q) factors through Ar (V), and pendent of the Weil cohomology (because this is true H num num I claim that the map of the irreducible factors of the numerator and de- r/2 nominator). If |ι(α)| = q for every eigenvalue α of a ⊗ f 7→ a f : Q ⊗ Ar (V) → Ar (V,Q) r num num π acting on H (V,Q) and embedding ι of Q(α) into C, r then the Pr(T) are relatively prime in pairs, and so the is an isomorphism. As Anum(V,Q) is spanned by the r Weil conjectures W1–W4 are true for V. image of Anum(V), the map is obviously surjective. Let r e1,...,em be a Q-basis for Anum(V), and let f1,..., fm be d−r m the dual basis in Anum(V). If ∑i=1 ai ⊗ei (ai ∈ Q) becomes Application to Rings of Correspondences r zero in Anum(V,Q), then a j = (∑aiei)· f j = 0 for all j. Thus We show that the (mere) existence of a Weil co- the map is injective. dim(V) homology theory implies that the Q-algebra of cor- Theorem 2.13. The Q-algebra Anum (V ×V) is finite- respondences for numerical equivalence on an alge- dimensional and semisimple. braic variety is a semisimple Q-algebra of finite di- Proof. Let d = dim(V) and B = Ad (V ×V). Then B is a mension. num finite-dimensional Q-algebra (2.11), and the pairing 2.8. Let V be a connected nonsingular projective variety of dimension d. An algebraic r-cycle on V is a f ,g 7→ ( f · g): B × B → Q

30 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 is nondegenerate. Let f be an element of the radical Weil cohomology in all dimensions!27 (Serre 2001, p. rad(B) of B. We have to show that ( f ·g) = 0 for all g ∈ B. 125, p. 255.) By the time Serre wrote up his seminar in d Let A = AH (V ×V,Q). Then A is a finite-dimensional September 1958, he was able to include a reference to Q-algebra, and there is a surjective homomorphism Grothendieck’s announcement (1958b) of a “Weil cohomology”. def d S d 2.12 A = AH (V ×V,Q) −→ Anum(V ×V,Q) ' Q ⊗ B. Grothendieck’s claim raised two questions:

As the ring A/rad(A) is semisimple, so also is its quo- (A) when G is commutative, is it possible to define tient (Q ⊗ B)/S(rad(A)). Therefore S(rad(A)) ⊃ rad(Q ⊗ B), higher cohomology groups? and so there exists an f 0 ∈ rad(A) mapping to 1⊗ f . For (B) assuming (A) are they the “true” cohomology all g ∈ A, groups when G is finite? (27) ( f 0 · gt ) = ∑(−1)i Tr( f 0 ◦ g | Hi(V)) To answer (A), Grothendieck observed that to define i a sheaf theory and a sheaf-cohomology, much less is needed than a topological space. In particular the —this can be proved exactly as (2.1). But f 0 ◦gt ∈ rad(A); “open subsets” need not be subsets. therefore it is nilpotent, and so its trace on Hi(V) is Specifically, let C be an essentially small category zero. Hence ( f 0 · gt ) = 0, and so (1 ⊗ f · S(gt )) = 0 for all admitting finite fibred products and a final object V, g ∈ A. It follows that f = 0. 2 and suppose that for each object U of C there is given Theorem 2.13 was extracted from Jannsen 1992. a family of “coverings” (Ui → U)i. The system of coverings is said to be a Grothendieck topology on C if it Etale Cohomology satisfies the following conditions: 0 From all the work of Grothendieck, it is without doubt étale (a) (base change) if (Ui → U) is a covering and U → 0 0 cohomology which has exercised the most profound influence U is a morphism in C, then (Ui ×U U → U ) is a on the development of arithmetic geometry in the last fifty covering; years. Illusie 2014.26 (b) (local nature) if (Ui → U)i is a covering, and, for each i, (Ui, j → Ui) j is a covering, then the family of With his definition of fibre spaces on algebraic composites (Ui, j → U)i, j is a covering; varieties, Weil began the process of introducing into (c) a family consisting of a single isomorphism abstract algebraic geometry the powerful topological ϕ : U0 → U is a covering. methods used in the study of complex algebraic varieties. He introduced fibre spaces in a 1949 confer- For example, let V be a topological space, and con- ence talk, and then, in more detail, in a course at the sider the category C whose objects are the open sub- University of Chicago (Weil 1952). For the first time, sets of V with the inclusions as morphisms; then the he made use of the Zariski topology in his definition coverings of open subsets in the usual sense define a of an abstract variety, and he equipped his varieties Grothendieck topology on C. with this topology. He required a fibre space to be lo- Consider a category C equipped with a Grothen- cally trival for the Zariski topology on the base vari- dieck topology. A presheaf is simply a contravariant ety. Weil’s theory works much as expected, but some functor from C to the category of abelian groups. A fibre spaces that one expects (from topology) to be lo- presheaf P is a sheaf if, for every covering (Ui → U)i, cally trivial are not, because the Zariski topology has the sequence too few open sets. In a seminar in April 1958, Serre enlarged the P(U) → ∏P(Ui) ⇒ ∏P(Ui ×U Uj) scope of Weil’s theory by admitting also fibre spaces i i, j that are only “locally isotrivial” in the following sense: is exact. With these definitions, the sheaf theory there exists a covering V = S U of the base variety V i i in Grothendieck 1957 carries over almost word-for- by open subvarieties U and finite étale maps U0 → U i i i word. The category of sheaves is abelian, satisfies such that the fibre space becomes trivial when pulled Grothendieck’s conditions (AB5) and (AB3*), and ad- back to each U0. For an algebraic group G over k, Serre i mits a family of generators. Therefore, it has enough defined H1(V,G) to be the set of isomorphism classes injectives, and the cohomology groups can be defined of principal fibre spaces on V under G, which he con- to be the right derived functors of F F(V). sidered to be the “good H1”. At the end of the semi- When Grothendieck defined the étale topology on nar, Grothendieck said to Serre that this will give the a variety (or scheme) V, he took as coverings those 26 De toute l’oeuvre de Grothendieck, c’est sans doute la coin Serre’s definition of “locally isotrivial”, but Mike homologie étale qui aura exercé l’influence la plus profonde sur l’évolution de la géométrie arithmétique dans les cin- 27 Dès la fin de l’exposé oral, Grothendieck m’a dit: cela va quante dernières années. donner la cohomologie de Weil en toute dimension!

DECEMBER 2016 NOTICES OF THE ICCM 31 Artin realized that it was better to allow as coverings a few years passed before this idea really took shape: all surjective families of étale morphisms. With his Grothendieck did not see how to start. He also had other occupations.32 (Illusie 2014, p. 175.) definition the local rings satisfy Hensel’s lemma. A test for (B) is: When Grothendieck came to Harvard in 1961, Mike

(C) let V be a nonsingular algebraic variety over C, and let Artin asked him: Λ be a finite abelian group; do the étale cohomology groups if it was all right if I thought about it, and so that was the Hr(V ,Λ) coincide with the singular cohomology groups et beginning… [Grothendieck] didn’t work on it until I proved Hr(V an,Λ)? the first theorem. … I thought about it that fall … And then For r = 0, this just says that an algebraic variety over I gave a seminar… (Segel 2009, p. 358.) C is connected for the complex topology if it is con- Serre writes (email July 2015): nected for the Zariski topology. For r = 1, it is the Riemann existence theorem,28 which says that every Grothendieck, after the seminar lecture where I defined 1 finite covering of V an is algebraic. In particular, (C) “the good H ”, had the idea that the higher cohomology groups would also be the good ones. But, as far as I know, is true for curves. It was probably this that made he could not prove their expected properties… It was Mike Grothendieck optimistic that (C) is true in all de- Artin, in his Harvard seminar notes of 1962, who really 1 grees. started the game, by going beyond H . For instance, he proved that a smooth space of dimension 2 minus a point Grothendieck thought always in relative terms: one space has (locally) the same cohomology as a 3-sphere (Artin 1962, over another. Once the cohomology of curves (over an al- p. 110). After that, he and Grothendieck took up, with SGA gebraically closed field) was understood, we could expect 4: la locomotive de Bures était lancée. similar results for the direct images for a relative curve …, and, “by unscrewing [dévissage]”… for the higher Hr.29 (Il- In 1963–1964, Artin and Grothendieck organized lusie 2014, p. 177.) their famous SGA 4 seminar. Initially (in 1958), Serre was less sure: The étale topology gives good cohomology groups only for torsion groups. To obtain a Weil co- 1 Of course, the Zariski topology gives a π1 and H that are too small, and I had fixed that defect. But was that enough? homology theory, it is necessary to define My reflexes as a topologist told me that we must also deal 30 r r n with the higher homotopy groups: π2, π3, etc. (Serre 2001, H (V , ) = H (V , /` ), et Z` ←−lim et Z Z p. 255.) n

Artin proved (C) by showing that, in the relative di- r and then tensor with Q` to get H (Vet,Q`). This does mension one case, the Zariski topology is sufficiently give a Weil cohomology theory, and so the Weil con- fine to give coverings by K(π,1)’s (SGA 4, XI).31 jectures (W1–W3) hold with Although Grothendieck had the idea for étale cohomology in 1958, r Pr(V,T) = det(1 − πT | H (Vet,Q`)). 28 Riemann used the Dirichlet principle (unproven at the time) to show that on every compact Riemann surface S there More generally, Grothendieck (1964) proved that, for are enough meromorphic functions to realize S as a projec- every algebraic variety V over a finite field k , tive algebraic curve; this proves the Riemann existence the- 0 0 orem for nonsingular projective curves. r+1 29 r (−1) Grothendieck pensait toujours en termes relatifs: un es- (28) Z(V0,T) = ∏det(1 − πT | Hc (Vet,Q`)) pace au-dessus d’un autre. Une fois la cohomologie des r courbes (sur un corps algébriquement clos) tirée au clair, on pouvait espérer des résultats similaires pour les images where Hc denotes cohomology with compact support. directes pour une courbe relative (les théorèmes de spécial- In the situation of (W5), isation du π1 devaient le lui suggérer), et, “par dévissage” (fibrations en courbes, suites spectrales de Leray), atteindre r r ˜ les Hi supérieurs. H (Vet,Q`) ' H (V(C),Q) ⊗ Q` 30 1 Bien sûr, la topologie de Zariski donne un π1 et un H trop petits, et j’avais remédié à ce défaut. Mais était-ce suffisant? (proper and smooth base change theorem), and so Mes réflexes de topologue me disaient qu’il fallait aussi s’oc- the `-adic Betti numbers of V are independent of `. If cuper des groupes d’homotopie supérieurs: π2, π3, etc. 31 This is rather Serre’s way of viewing Artin’s proof. As Serre the Riemann hypothesis holds, then they equal Weil’s wrote (email July 2015): Betti numbers. In the years since it was defined, étale cohomol- [This] was the way I saw it, and I liked it for two reasons: a) it is a kind of explanation why higher homotopy groups ogy has become such a fundamental tool that today’s don’t matter: they don’t occur in these nice Artin neighbour- arithmetic geometers have trouble imagining an age hoods; b) the fundamental group of such a neighbourhood in which it didn’t exist. has roughly the same structure (iterated extension of free groups) as the braid groups which were so dear to Emil Artin; 32 in particular, it is what I called a “good group”: its cohomol- Quelques années s’écoulèrent avant que cette idée ne ogy is the same when it is viewed as a discrete group or as a prenne réellement forme: Grothendieck ne voyait pas com- profinite group. ment démarrer. Il avait aussi d’autres occupations.

32 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 de Rham Cohomology (Characteristic Zero) would have given only the H0,r part of the cohomology). Let V be a nonsingular algebraic variety over a In 1966, Grothendieck discovered how to obtain field k. Define H∗ (V) to be the (hyper)cohomology of dR the de Rham cohomology groups in characteristic the complex zero without using differentials, and suggested that • d 1 d d r d his method could be modified to give a good p-adic ΩV/k = OX −→ ΩV/k −→··· −→ ΩV/k −→··· cohomology theory in characteristic p. of sheaves for the Zariski topology on V. When k = C, Let V be a nonsingular variety over a field k of ∗ an • we can also define HdR(V ) by replacing ΩV/k with the characteristic zero. Define inf(V/k) to be the category complex of sheaves of holomorphic differentials on whose objects are open subsets U of V together with V an for the complex topology. Then thickening of U, i.e., an immersion U ,→ T defined by a nilpotent ideal in OT . Define a covering fam- r an r an H (V ,Q) ⊗Q C ' HdR(V ) ily of an object (U,U ,→ T) of inf(V/k) to be a family (Ui,Ui ,→ Ti)i with (Ti)i a Zariski open covering of T for all r. and Ui = U ×T Ti. These coverings define the “infinites- When k = , there is a canonical homomorphism C imal” Grothendieck topology on inf(V/k). There is a ∗ ∗ an structure sheaf OV on Vinf, and Grothendieck proves HdR(V) → HdR(V ). inf that In a letter to Atiyah in 1963, Grothendieck proved H∗(V ,O ) ' H∗ (V) that this is an isomorphism (Grothendieck 1966). inf Vinf dR Thus, for a nonsingular algebraic variety over a field k (Grothendieck 1968, 4.1). of characteristic zero, there are algebraically defined This doesn’t work in characteristic p, but Grot- ∗ cohomology groups HdR(V) such that, for every em- hendieck suggested that by adding divided powers to bedding ρ : k → C, the thickenings, one should obtain a good cohomology in characteristic p. There were technical problems ∗ ∗ ∗ an HdR(V) ⊗k,ρ C ' HdR(ρV) ' HdR((ρV) ). at the prime 2, but Berthelot resolved these in this thesis to give a good definition of the “crystalline” This gives a Weil cohomology theory with coefficients site, and he developed a comprehensive treatment in k. of crystalline cohomology (Berthelot 1974). This is a cohomology theory with coefficients in the ring of p-Adic Cohomology Witt vectors over the base field k. On tensoring it The étale topology gives Weil cohomologies with with the field of fractions, we obtain a Weil cohomology. coefficients in Q` for all primes ` different from the characteristic of k. Dwork’s early result (see p. 28) About 1975, Bloch extended Serre’s sheaf WOV to suggested that there should also be p-adic Weil co- a “de Rham-Witt complex” homology theories, i.e., a cohomology theories with d d WΩ• : W −→ WΩ1 −→··· coefficients in a field containing Qp. V/k OV V/k Let V be an algebraic variety over a field k of characteristic p 6= 0. When Serre defined the cohomology and showed that (except for some small p) the Zariski groups of coherent sheaves on algebraic varieties, he cohomology of this complex is canonically isomor- asked whether the formula phic to crystalline cohomology (Bloch 1977). Bloch • used K-theory to define WΩV/k (he was interested in ? j i βr(V) = ∑ dimk H (V,ΩV/k) relating K-theory to crystalline cohomology among i+ j=r other things). Deligne suggested a simpler, more direct, definition of the de Rham-Witt complex and this gives the “true” Betti numbers, namely, those inter- approach was developed in detail by Illusie and Ray- vening in the Weil conjectures (Serre 1954, p. 520). An example of Igusa (1955) showed that this for- naud (Illusie 1983). mula gives (at best) an upper bound for Weil’s Betti Although p-adic cohomology is more difficult to L j i define than `-adic cohomology, it is often easier to numbers. Of course, the groups i+ j=r H (V,ΩV/k) wouldn’t give a Weil cohomology theory because the compute with it. It is essential for understanding coefficient field has characteristic p. Serre (1958) p-phenomena, for example, p-torsion, in characteris- next considered the Zariski cohomology groups tic p. r H (V,WOV ) where WOV is the sheaf of Witt vec- Notes. The above account of the origins of p-adic co- tors over OV (a ring of characteristic zero), but homology is too brief—there were other approaches found that they did not have good properties (they and other contributors.

DECEMBER 2016 NOTICES OF THE ICCM 33 3. The Standard Conjectures Hence, q−1/2 f is a unitary operator on H1,0(V), and so its eigenvalues a ,...,a have absolute value 1. Alongside the problem of resolution of singularities, the 1 g ∗ 1 1/2 1/2 proof of the standard conjectures seems to me to be the most The eigenvalues of f on H (V,Q) are q a1,...,q ag, 1/2 1/2 1/2 urgent task in algebraic geometry. q a¯1,...,q a¯g, and so they have absolute value q . Grothendieck 1969. In higher dimensions, an extra condition is cer- We have seen how to deduce the first three of tainly needed (consider a product). Serre realized that the Weil conjectures from the existence of a Weil co- it was necessary to introduce a polarization. homology. What more is needed to deduce the Rie- Theorem 3.1 (Serre 1960, Thm 1). Let V be a con- mann hypothesis? About 1964, Bombieri and Grot- nected nonsingular projective variety of dimension d hendieck independently found the answer: we need a Künneth formula and a Hodge index theorem for over C, and let f be an endomorphism V. Suppose that algebraic classes. Before explaining this, we look at there exists an integer q > 0 and a hyperplane section E of V such that f −1(E) is algebraically equivalent to the analogous question over C. qE. Then, for all r ≥ 0, the eigenvalues of f acting on Hr(V, ) have absolute value qr/2. A Kählerian Analogue Q For a variety over a finite field and f the Frobe- In his 1954 ICM talk, Weil sketched a transcen- nius map, f −1(E) is equivalent to qE, and so this is dental proof of the inequality σ(ξ ◦ ξ 0) > 0 for correspondences on a complex curve, and wrote: truly a kählerian analogue of the Riemann hypothesis over finite fields. Note that the condition f −1(E) ∼ qE … this is precisely how I first persuaded myself of the implies that ( f −1Ed) = qd(Ed), and hence that f has de- truth of the abstract theorem even before I had perceived d the connection between the trace σ and Castelnuovo’s equiv- gree q . alence defect. The proof is an application of two famous the- In a letter to Weil in 1959, Serre wrote: orems. Throughout, V is as in the statement of the theorem. Let E be an ample divisor on V, let u be its In fact, a similar process, based on Hodge theory, applies class in H2(V, ), and let L be the “Lefschetz operator” to varieties of any dimension, and one obtains both the posi- Q tivity of certain traces, and the determination of the absolute ∗ ∗+2 values of certain eigenvalues in perfect analogy with your x 7→ u ∪ x: H (V,Q) → H (V,Q)(1). beloved conjectures on the zeta functions.33 We now explain this. More concretely, we consider the Theorem 3.2 (Hard Lefschetz). For r ≤ d, the map following problem: Let V be a connected nonsingu- Ld−r : Hr(V, ) → H2d−r(V, )(d − r) lar projective variety of dimension d over C, and let Q Q f : V → V be an endomorphism of degree qd; find conditions on f ensuring that the eigenvalues of f acting is an isomorphism. r r/2 on H (V,Q) have absolute value q for all r. Proof. It suffices to prove this after tensoring with For a curve V, no conditions are needed. The ac- C. Lefschetz’s original “topological proof” (1924) is tion of f on the cohomology of V preserves the Hodge inadequate, but there are analytic proofs (e.g., Weil decomposition 1958, IV, n◦6, Cor. to Thm 5). 2 1 1,0 0,1 i, j def j i H (V, ) ' H (V) ⊕ H (V), H (V) = H (V,Ω ), r r C Now let H (V) = H (V,C), and omit the Tate twists. , Suppose that r ≤ d, and consider and the projection H1(V,C) → H1 0(V) realizes 1 1 1,0 H (V,Z) ⊂ H (V,C) as a lattice in H (V), stable under d−r 1,0 r−2 L r L 2d−r L 2d−r+2 the action of f . Define a hermitian form on H (V) by H (V) H (V) ' H (V) H (V). 1 Z hω,ω0i = ω ∧ ω¯ 0. The composite of the maps is an isomorphism (3.2), 2πi V and so This is positive definite. As f ∗ acts on H2 (V) as mul- dR Hr(V) = Pr(V) ⊕ LHr−2(V) tiplication by deg( f ) = q, d−r+1 r r L 2d−r+2 def 1 Z q Z def with P (V) = Ker(H (V) −−−−→ H (V)). On repeat- h f ∗ω, f ∗ω0i = f ∗(ω ∧ ω¯ 0) = ω ∧ ω¯ 0 = qhω,ω0i. 2πi V 2πi V ing this argument, we obtain the first of the following decompositions, and the second is proved similarly: 33 “En fait, un procédé analogue, basé sur la théorie de Hodge, s’applique aux variétés de dimension quelconque, et M L jPr−2 j if r ≤ d l’on obtient à la fois la positivité de certaines traces, et la  détermination des valeurs absolues de certaines valeurs pro- r  j≥0 H (V) = M pres, en parfaite analogie avec tes chères conjectures sur les L jPr−2 j if r ≥ d.  fonctions zêta.”  j≥r−d

34 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 In other words, every element x of Hr(V) has a unique Proof. Exercise, using (3.4). See Weil 1958, IV, n◦7, expression as a sum Thm 8. 2

j r−2 j To deduce (1.2) from the theorem, it is necessary (29) x = ∑ L x j, x j ∈ P (V). j≥max(r−d,0) to show that the nonalgebraic cycles contribute only positive terms. This is what Hodge did in his 1937 The cohomology classes in Pr(V), r ≤ d, are said to be paper. primitive. We now prove Theorem 3.1. It follows from (3.4) The Weil operator C : H∗(V) → H∗(V) is the linear that the sesquilinear form map such that Cx = ia−bx if x is of type (a,b). It is an ∗ r r automorphism of H (V) as a C-algebra, and C2 acts (30) (x,y) 7→ A(x,Cy¯): H (V) × H (V) → C on Hr(V) as multiplication by (−1)r (Weil 1958, n◦5, is hermitian and positive definite. Let g = q−r/2Hr( f ). p. 74). r Then g respects the structure of H∗(V) as a bigraded Using the decomposition (29), we define an oper- r -algebra, the form I, and the operators a 7→ a¯ and ator ∗: Hr(V) → H2d−r(V) by C a 7→ La. Therefore, it respects the form (30), i.e., it is a r(r+1) unitary operator, and so its eigenvalues have absolute 2 j! d−r− j ∗x = ∑ (−1) · (d−r− j)! ·C(L x j). j≥max(r−d,0) value 1. This completes the proof of the theorem. This proof extends to correspondences. For For ω ∈ Hr(V), let curves, it then becomes the argument in Weil’s ICM talk; for higher dimensions, it becomes the proof of R ω if r = 2d I(ω) = V Theorem 2 of Serre 1960. 0 if r < 2d, and let Weil Forms I(x,y) = I(x · y). As there is no Weil cohomology theory in nonzero characteristic with coefficients in a real field, it is not ∗ Lemma 3.3. For x,y ∈ H (V), possible to realize the Frobenius map as a unitary operator. Instead, we go back to Weil’s original idea. I(x,∗y) = I(y,∗x) and I(x,∗x) > 0 if x 6= 0. Let H be a Weil cohomology theory over k with Proof. Weil 1958, IV, n◦7, Thm 7. 2 coefficient field Q. From the Künneth formula and Poincaré duality, we obtain isomorphisms j j For x = ∑L x j and y = ∑L y j with x j, y j primitive of degree r − 2 j, put 2d H2d(V ×V)(d) ' M Hr(V) ⊗ H2d−r(V)(d) A(x,y) r=0 2d r(r+1) j! j d−r+2 j M r 2 ' EndQ-linear(H (V)). = ∑ (−1) · (d−r− j)! · (−1) · I(u · x j · y j). j≥max(r−d,0) r=0

Theorem 3.4. The map A is a bilinear form on Hr(V), Let πr be the rth Künneth projector. Under the isomor- and phism, the subring H2d(V ×V) def= ◦ H2d(V ×V) ◦ A(y,x) = (−1)rA(x,y), A(Cx,Cy) = A(x,y) r πr πr 2d r A(x,Cy) = A(y,Cx), A(x,Cx¯) > 0 if x 6= 0. of H (V ×V)(d) corresponds to EndQ-linear(H (V)). r 2r Let AH (−) denote the Q-subspace of H (−)(r) gen- Proof. The first two statements are obvious, and the d erated by the algebraic classes. Then AH (V ×V) is the second two follow from (3.3) because Q-algebra of correspondences on V for homological equivalence (see 2.9). Assume that the Künneth pro- A(x,Cy) = I(L jx ,∗L jy ). ∑ j j jectors are algebraic, and let j≥max(r−d,0) πr d d 2d d See Weil 1958, IV, n◦7, p. 78. 2 AH (V ×V)r = AH (V ×V)∩H (V ×V)r = πr ◦AH (V ×V)◦πr. Note that the intersection form I on Hd(V) is sym- Then metric or skew-symmetric according as d is even or 2d d M d odd. AH (V ×V) = AH (V ×V)r. r=0 Theorem 3.5 (Hodge Index). Assume that the dimen- Let sion d of V is even. Then the signature of the intersec- d a a,b r r tion form on H (V) is ∑a,b(−1) h (V). φ : H (V) × H (V) → Q(−r)

DECEMBER 2016 NOTICES OF THE ICCM 35 be a nondegenerate bilinear form. For α ∈ End(Hr(V)), be the Lefschetz operator x 7→ u·x defined by the class we let α0 denote the adjoint of α with respect to φ: u of an ample divisor in H2(V)(1); then, for all r ≤ d, the map φ(αx,y) = φ(x,α0y). Ld−r : Hr(V) → H2d−r(V)(d − r) Definition 3.6. We call φ a Weil form if it satisfies the is an isomorphism. As before, this gives decomposi- following conditions: tions (a) φ is symmetric or skew-symmetric according as r r M j r−2 j is even or odd; H (V) = L P (V) d 0 d j≥max(r−d,0) (b) for all α ∈ AH (V ×V)r, the adjoint α ∈ AH (V ×V)r; 0 0 moreover, Tr(α ◦α ) ∈ Q, and Tr(α ◦α ) > 0 if α 6= 0. with Pr(V) equal to the kernel of Ld−r+1 : Hr(V) → 2d−r+2 r Example 3.7. Let V be a nonsingular projective vari- H (V). Hence x ∈ H (V) has a unique expression as a sum ety over C. For all r ≥ 0, the pairing r r (31) x = L jx , x ∈ Pr−2 j(V). H (V) × H (V) → C, x,y 7→ (x · ∗y) ∑ j j j≥max(r−d,0) is a Weil form (Serre 1960, Thm 2). Define an operator Λ: Hr(V) → Hr−2(V) by Example 3.8. Let C be a curve over k, and let J be its j−1 jacobian. The Weil pairing φ : T`J ×T`J → T`Gm extends Λx = ∑ L x j. by linearity to a Q`-bilinear form j≥max(r−d,0)

H1(Cet,Q`) × H1(Cet,Q`) → Q`(1) The Standard Conjectures of Lefschetz Type A(V,L): For all r ≤ d, the isomorphism Ld−2r on H (C , ) = ⊗T J. This is Weil form (Weil 1948b, 2 : 1 et Q` Q` ` 2r 2d−2r VI, n◦48, Thm 25). H (V)(r) → H (V)(d −r) restricts to an isomorphism Proposition 3.9. If there exists a Weil form on Hr(V), r d−r d AH (V) → AH (V). then the Q-algebra AH (V ×V)r is semisimple. 2r d−2r Proof. It admits a positive involution α 7→ α0, and so Equivalently, x ∈ H (V)(r) is algebraic if L x is we can argue as in the proof of (1.28). 2 algebraic. B(V): The operator Λ is algebraic, i.e., it lies in the im- r Proposition 3.10. Let φ be a Weil form on H (V), and age of d 0 let α be an element of A (V ×V)r such that α ◦ α is an integer q. For every homomorphism ρ : [α] → , d−1 2d−2 Q C AH (V ×V) → H (V ×V)(d − 1) M r+2 r ρ(α0) = ρ(α) and |ρα| = q1/2. ' Hom(H (V),H (V)). r≥0 Proof. The proof is the same as that of (1.29). 2 C(V): The Künneth projectors πr are algebraic. Equiv- alently, the Künneth isomorphism The Standard Conjectures ∗ ∗ ∗ Roughly speaking, the standard conjectures state H (V ×V) ' H (V) ⊗ H (V) that the groups of algebraic cycles modulo homologi- induces an isomorphism cal equivalence behave like the cohomology groups of a Kähler manifold. Our exposition in this subsection A∗ (V ×V) ' A∗ (V) ⊗ A∗ (V). follows Grothendieck 1969 and Kleiman 1968.34 H H H Let H be a Weil cohomology theory over k with Proposition 3.11. There are the following relations coefficient field Q. We assume that the hard Lefschetz among the conjectures. theorem holds for H.35 This means the following: let L (a) Conjecture A(V ×V,L ⊗ 1 + 1 ⊗ L) implies B(V). 34 According to Illusie (2010): Grothendieck gave a series of (b) If B(V) holds for one choice of L, then it holds for lectures on motives at the IHÉS. One part was about the stan- all. dard conjectures. He asked John Coates to write down notes. Coates did it, but the same thing happened: they were re- (c) Conjecture B(V) implies A(V,L) (all L) and C(V). turned to him with many corrections. Coates was discour- aged and quit. Eventually, it was Kleiman who wrote down Proof. Kleiman 1994, Theorem 4-1. 2 the notes in Dix exposés sur la cohomologie des schémas. Thus A(V,L) holds for all V and L if and only if 35 For `-adic étale cohomology, this was proved by Deligne as a consequence of his proof of the Weil conjectures, and it B(V) holds for all V; moreover, each conjecture im- follows for the other standard Weil cohomology theories. plies Conjecture C.

36 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Example 3.12. Let k = F. A smooth projective variety Theorem 3.15. Assume the standard conjectures. V over k arises from a variety V0 defined over a finite Then subfield k0 of k. Let π be the corresponding Frobenius r r r H (V) × H (V) → Q, x,y 7→ (x · ∗y), endomorphism of V, and let Pr(T) = det(1−πT | H (V)). According to the Cayley-Hamilton theorem, Pr(π) acts is a Weil form. r as zero on H (V). Assume that the Pr are relatively Proof. Kleiman 1968, 3.11. 2 prime (this is true, for example, if the Riemann hy- d pothesis holds). According to the Chinese remainder Corollary 3.16. The Q-algebra AH (V ×V)r is semisim- theorem, there are polynomials Pr(T) ∈ Q[T] such that ple.

1 mod P (T) Proof. It admits a positive involution; see (2.4). 2 Pr(T) = r 0 mod P (T) for s 6= r. s Theorem 3.17. Assume the standard conjectures. Let Now Pr(π) projects H∗(V) onto Hr(V), and so Conjec- V be a connected nonsingular projective variety of di- ture C(V) is true. mension d over k, and let f be an endomorphism V. Suppose that there exists an integer q > 0 and a hyper- Example 3.13. Conjecture B(V) holds if V is an plane section E of V such that f −1(E) is algebraically abelian variety or a surface with dimH1(V) equal to equivalent to qE. Then, for all r ≥ 0, the eigenvalues of twice the dimension of the Picard variety of V f acting on Hr(V, ) have absolute value qr/2. (Kleiman 1968, 2. Appendix). Q Proof. Use E to define the Lefschetz operator. Then These are essentially the only cases where the 0 standard conjectures of Lefschetz type are known α ◦ α = q, and we can apply (3.10). 2 (see Kleiman 1994). Aside 3.18. In particular, the standard conjec- The Standard Conjecture of Hodge Type tures imply that the Frobenius endomorphism acts semisimply on étale cohomology over . For abelian r F For r ≤ d, let AH (V)pr denote the “primitive” part varieties, this was proved by Weil in the 1940s, but Ar (V)∩P2r(V) of Ar (V). Conjecture A(V,L) implies that H H there has been almost no progress since then. r M j r− j AH (V) = L A (V)pr. j≥max(2r−d,0) The Standard Conjectures and Equivalences on I(V,L): For r ≤ d, the symmetric bilinear form Algebraic Cycles

r d−2r r r Our statement of the standard conjectures is rel- x,y 7→ (−1) (x · y · u ): AH (V)pr × AH (V)pr → Q ative to a choice of a Weil cohomology theory. Grot- is positive definite. hendieck initially stated the standard conjectures in a letter to Serre37 for algebraic cycles modulo alge- In characteristic zero, the standard conjecture of braic equivalence, but an example of Griffiths shows Hodge type follows from Hodge theory (see 3.3). In nonzero characteristic, almost nothing is known ex- that they are false in that context. 0 cept for surfaces where it becomes Theorem 1.2. Recall that two algebraic cycles Z and Z on a variety V are rationally equivalent if there exists an alge- Consequences of the Standard Conjectures 1 0 braic cycle Z on V × P such that Z 0 = Z and Z 1 = Z ; Proposition 3.14. Assume the standard conjectures. that they are algebraically equivalent if there exists a r d−r For every x ∈ AH (V), there exists a y ∈ AH (V) such that curve T and an algebraic cycle Z on V × T such that 0 x · y 6= 0. Z t0 = Z and Z t1 = Z for two points t0,t1 ∈ T(k); that j they are homologically equivalent relative to some Proof. We may suppose that x = L x j with x j ∈ r− j fixed Weil cohomology H if they have the same class A (V)pr. Now H in H2∗(V)(∗); and that they are numerically equivalent j j d−2r d−2r+2 j 0 (L x j · L x j · u ) = (x j · x j · u ) > 0. 2 if (Z ·Y) = (Z ·Y) for all algebraic cycles Y of complementary dimension. We have Using the decomposition (31), we define an operator ∗: Hr(V) → H2d−r(V) by36 rat =⇒ alg =⇒ hom =⇒ num.

(r−2 j)(r−2 j+1) d−r+ j ∗x = ∑ (−1) 2 L (x j). For divisors, rational equivalence coincides with j≥max(r−d,0) linear equivalence. Rational equivalence certainly differs from algebraic equivalence, except over the alge- 36 This differs from the definition in kählerian geometry by some scalar factors. Over C, our form x,y 7→ (x·∗y) is positive braic closure of a finite fields, where all four equiva- definite on some direct summands of Hr(V) and negative def- lence relations are conjectured to coincide (folklore). inite on others, but this suffices to imply that the involution α 7→ α0 is positive. 37 Grothendieck-Serre Correspondence, p. 232, 1965.

DECEMBER 2016 NOTICES OF THE ICCM 37 For divisors, algebraic equivalence coincides with 3.21. The Hodge conjecture implies the standard con- numerical equivalence (Matsusaka 1957). For many jecture of Lefschetz type over C (obviously). Con- decades, it was believed that algebraic equivalence versely, the standard conjecture of Lefschetz type over and numerical equivalence coincide—one of Sev- C implies the Hodge conjecture for abelian varieties eri’s “self-evident” postulates even has this as a (Abdulali 1994, André 1996), and hence the standard conjecture of Hodge type for abelian varieties in all consequence (Brigaglia et al. 2004, p. 327). characteristics (Milne 2002). Griffiths (1969) surprised everyone by showing that, even in the classical situation, algebraic equiv- 3.22. Tate’s conjecture implies the standard conjec- alence differs from homological equivalence.38 How- ture of Lefschetz type (obviously). If the full Tate con- ever, there being no counterexample, it remains a jecture is true over finite fields of characteristic p, then the standard conjecture of Hodge type holds in char- folklore conjecture that numerical equivalence coin- acteristic p. cides with homological equivalence for the standard Weil cohomologies. For `-adic étale cohomology, this Here is a sketch of the proof of the last statement. conjecture is stated in Tate 1964. Let k denote an algebraic closure of Fp. The author The standard conjectures for a Weil cohomol- showed that the Hodge conjecture for CM abelian ogy theory H imply that numerical equivalence coin- varieties over C implies the standard conjecture of cides with homological equivalence for H (see 3.14). It Hodge type for abelian varieties over k. The proof uses would be useful to have a statement of the standard only that Hodge classes on CM abelian varieties are conjectures independent of any Weil cohomology the- almost-algebraic at p (see (4.5) below for this notion). The Tate conjecture for finite subfields of k implies ory. One possibility is to state them for the Q-vector spaces of algebraic cycles modulo smash-nilpotent the standard conjecture of Lefschetz type over k (ob- equivalence in the sense of Voevodsky 1995 (which viously), and, using ideas of Abdulali and André, one implies homological equivalence, and is conjectured can deduce that Hodge classes on CM abelian vari- to equal numerical equivalence). eties are almost-algebraic at p; therefore the standard conjecture of Hodge type holds for abelian varieties over k. The full Tate conjecture implies that the cat- The Standard Conjectures and the Conjectures of egory of motives over k is generated by the motives Hodge and Tate of abelian varieties, and so the Hodge standard conjecture holds for all nonsingular projective varieties Grothendieck hoped that his standard conjec- over k. A specialization argument now proves it for tures would be more accessible than the conjectures all varieties in characteristic p. of Hodge and Tate, but these conjectures appear to be closely intertwined. Before explaining this, I recall the statements of the Hodge and Tate conjec- 4. Motives tures. For Grothendieck, the theory of “motifs” took on Conjecture 3.19 (Hodge). Let V be a smooth projec- an almost mystical meaning as the reality that lay be- tive variety over C. For all r ≥ 0, the Q-subspace of neath the “shimmering ambiguous surface of things”. r r H2 (V,Q) spanned by the algebraic classes is H2 (V,Q)∩ But in their simplest form, as a universal Weil coho- Hr,r(V). mology theory, the idea is easy to explain. Having already written a “popular” article on motives (Milne Conjecture 3.20 (Tate). Let V0 be a smooth projective 2009b), I shall be brief. variety over a field k0 finitely generated over its prime Let ∼ be an adequate equivalence relation on al- field, and let ` be a prime 6= char(k). For all r ≥ 0, the gebraic cycles, for example, one of those listed on 2r Q`-subspace of H (Vet,Q`(r)) spanned by the algebraic p. 37. We want to define a Weil cohomology theory classes consists exactly of those fixed by the action of that is universal among those for which ∼⇒ hom. We Gal(k/k ). Here k is a separable algebraic closure of k 0 0 also want to have Q as the field of coefficients, but and V = (V0)k. we know that this is impossible if we require the tar- By the full Tate conjecture, I mean the Tate con- get category to be that of Q-vector spaces, and so we jecture plus num = hom(`) (cf. Tate 1994, 2.9). only ask that the target category be Q-linear and have certain other good properties. 38 In fact, they aren’t even close. The first possible counterex- Fix a field k and an adequate equivalence rela- ample is for 1-cycles on a 3-fold, and, indeed, for a complex tion ∼. The category Corr∼(k) of correspondences over algebraic variety V of dimension 3, the vector space k has one object hV for each nonsingular projective { one − cycles homologically equivalent to zero } variety over k; its morphisms are defined by ⊗ { one − cycles algebraically equivalent to zero } Q M M Hom(hV,hW) = Homr(hV,hW) = Adim(V)+r(V ×W). may be infinite dimensional (Clemens 1983). ∼ r r

38 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 The bilinear map in (2.9) can now be written hV ⊗ hW = h(V ×W).

r s r+s Hom (V1,V2) × Hom (V2,V3) → Hom (V1,V3), Both of these structures extend to the category of motives. The functor − ⊗ h2(P1) is h(V,e,m) 7→ h(V,e,m + 1), and its associativity means that Corr∼(k) is a cate- which is an equivalence of categories (because we al- gory. Let H be a Weil cohomology theory. An element low negative m). of Homr(hV,hW) defines a homomorphism H∗(V) → H∗+r(W) of degree r. In order to have the gradations 0 Properties of the Category of Motives preserved, we consider the category Corr∼(k) of correspondences of degree 0, i.e., now For the language of tensor categories, we refer the

0 dimV reader to Deligne and Milne 1982. Hom(hV,hW) = Hom (hV,hW) = A∼ (V ×W). 4.1. The category of motives is a Q-linear pseudo- Every Weil cohomology theory such that ∼⇒ hom fac- abelian category. If ∼= num, it is semisimple abelian. 0 If ∼= rat and k is not algebraic over a finite field, then tors uniquely through Corr∼(k) as a functor to graded vector spaces. it is not abelian. However, Corr0 (k) is not an abelian category. ∼ To say that M ∼(k) is Q-linear just means that There being no good notion of an abelian envelope of the Hom sets are Q-vector spaces and composition is a category, we define eff(k) to be the pseudo-abelian M ∼ -bilinear. Thus M ∼(k) is -linear and pseudo-abelian 0 Q Q envelope of Corr∼(k). This has objects h(V,e) with V as by definition. If ∼= num, the Hom sets are finite- dim(V) before and e an idempotent in the ring A∼ (V ×V); dimensional Q-vector spaces, and the Q-algebras its morphisms are defined by End(M) are semisimple (2.13). A pseudo-abelian category with these properties is a semisimple abelian Hom(h(V,e),h(W, f )) = f ◦ Hom(hV,hW) ◦ e. category (i.e., an abelian category such that every object is a direct sum of simple objects). For the last This is a pseudo-abelian category, i.e., idempotent en- statement, see Scholl 1994, 3.1. domorphisms have kernels and cokernels. The functor 4.2. The category of motives is a rigid tensor category.

0 eff If V is nonsingular projective variety of dimension d, hV h(V,id): Corr∼(k) → M∼ (k) then there is a (Poincaré) duality fully faithful and universal among functors from ∨ 0 h(V) ' h(V)(d). Corr∼(k) to pseudo-abelian categories. Therefore, every Weil cohomology theory such that ∼⇒ hom factors A tensor category is a symmetric monoidal cate- eff uniquely through M∼ (k). gory. This means that every finite (unordered) family eff The category M ∼ (k) is the category of effective of objects has a tensor product, well defined up to a motives over k. It is a useful category, but we need to given isomorphism. It is rigid if every object X admits enlarge it in order to have duals. One of the axioms a dual object X∨. The proof of (4.2) can be found in 2 1 for a Weil cohomology theory requires H (P ) to have Saavedra Rivano 1972, VI, 4.1.3.5. dimension 1. This means that the object H2(P1) is in- 4.3. Assume that the standard conjecture C holds for vertible in the category of vector spaces, i.e., the func- 2 1 some Weil cohomology theory. tor W H (P )⊗Q W is an equivalence of categories. In eff 1 (a) The category M (k) is abelian if and only if ∼= M ∼ (k), the object hP decomposes into a direct sum ∼ num. 1 0 1 2 1 hP = h P ⊕ h P , (b) The category M num(k) is a graded tannakian category. and we formally invert h2 1. When we do this, we P (c) The Weil cohomology theories such that hom= obtain a category M (k) whose objects are triples ∼ num correspond to fibre functors on the Tan- h(V,e,m) with h(V,e) as before and m ∈ Z. Morphisms nakian category M num(k). are defined by (a) We have seen (2.13) that M ∼(k) is abelian if dim(V)+n−m Hom(h(V,e,m),h(W, f ,n)) = f ◦ A∼ (V,W) ◦ e. ∼= num; the converse is proved in André 1996, Ap- pendice. This is the category of motives over k. It contains the (b) Let V be a nonsingular projective variety over category of effective motives as a full subcategory. k. Let π0,...,π2d be the images of the Künneth projec- 0 The category Corr∼(k) has direct sums and tensor d tors in Anum(V ×V). We define a gradation on 6 M num(k) products: by setting

r hV ⊕ hW = h(V tW) h(V,e,m) = h(V,eπr+2m,m).

DECEMBER 2016 NOTICES OF THE ICCM 39 Now the commutativity constraint can be modified so with this definition is that it depends on the choice that of σ. To remedy this, Deligne defines a cohomology class on V to be absolutely Hodge if it is Hodge relative dim(h(V,e,m)) = dim (eHr(V)). ∑ Q to every embedding . The problem with this defini- r≥0 σ tion is that, a priori, we know little more about abso- (rather than the alternating sum). Thus dim(M) ≥ 0 for lute Hodge classes than we do about algebraic classes. all motives M, which implies that M num(k) is tannakian To give substance to his theory, Deligne proved that (Deligne 1990, 7.1). all relative Hodge classes on abelian varieties are ab- (c) Let ω be a fibre functor on M num(k). Then solute. Hence they satisfy the standard conjectures V ω(hV) is a Weil cohomology theory such that and the Hodge conjecture, and so we have a theory of hom=num, and every such cohomology theory arises abelian motives over fields of characteristic zero that in this way. is much as Grothendieck envisaged it (Deligne 1982).

4.4. The standard conjectures imply that M num(k) has However, as Deligne points out, his theory works a canonical polarization. only in characteristic zero, which limits its useful- ness for arithmetic questions. Let A be an abelian The notion of a Weil form can be defined in any variety over the algebraic closure k of a finite field, tannakian category over . For example, a Weil form Q and lift A in two different ways to abelian varieties on a motive M of weight r is a map A1 and A2 in characteristic zero; let γ1 and γ2 be absolute Hodge classes on A and A of complementary φ : M ⊗ M → 1(−r) 1 2 dimension; then γ1 and γ2 define l-adic cohomology with the correct parity such that the map sending an classes on A for all primes l, and hence intersection endomorphism of M to its adjoint is a positive invo- numbers (γ1 · γ2)l ∈ Ql. The Hodge conjecture implies that (γ · γ ) lies in and is independent of l, but this lution on the Q-algebra End(M). To give a polarization 1 2 l Q is not known. on M num(k) is to give a set of Weil forms (said to be positive for the polarization) for each motive satisfy- However, the author has defined the notion of a ing certain compatibility conditions; for example, if φ “good theory of rational Tate classes” on abelian vari- and φ 0 are positive, then so also are φ ⊕ φ 0 and φ ⊗ φ 0. eties over finite fields, which would extend Deligne’s The standard conjectures (especially of Hodge type) theory to mixed characteristic. It is known that there exists at most one such theory and, if it exists, the imply that M num(k) admits a polarization for which the Weil forms defined by ample divisors are positive. rational Tate classes it gives satisfy the standard con- 0 0 jectures and the Tate conjecture; thus, if it exists, we Let 1 denote the identity object h (P ) in M ∼(k). Then, almost by definition, would have a theory of abelian motives in mixed characteristic that is much as Grothendieck envisaged it r 2r (Milne 2009a). AH (V) ' Hom(1,h (V)(r)).

r Almost-Algebraic Classes Therefore V h (V) has the properties expected of an “abstract” Weil cohomology theory. 4.5. Let V be an algebraic variety over an algebraically closed field k of characteristic zero. An almost- algebraic class on V of codimension r is an absolute Alternatives to the Hodge, Tate, and Standard Hodge class γ on V of codimension r such that there Conjectures exists a cartesian square

In view of the absence of progress on the Hodge, V V Tate, or standard conjectures since they were stated f more than fifty years ago, Deligne has suggested that, rather than attempting to proving these conjec- S Speck tures, we should look for a good theory of motives, and a global section ˜ of R2r f (r) satisfying the fol- based on “algebraically-defined”, but not necessarily γ ∗A lowing conditions (cf. Tate 1994, p. 76), algebraic, cycles. I discuss two possibilities for such algebraically-defined of cycles. • S is the spectrum of a regular integral domain of finite type over Z; Absolute Hodge Classes and Rational Tate Classes • f is smooth and projective; Consider an algebraic variety V over an alge- • the fibre of γ˜ over Spec(k) is γ, and the reduction braically closed field k of characteristic zero. If k is not of γ˜ at s is algebraic for all closed points s in a too big, then we can choose an embedding σ : k → C of dense open subset U of S. k in C and define a cohomology class on k to be Hodge If the above data can be chosen so that (p) is in the relative to σ if it becomes Hodge on σV. The problem image of the natural map U → SpecZ, then we say that

40 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 γ is almost-algebraic at p—in particular, this means case,40 but no “dévissage” worked for him.41 After that V has good reduction at p. he announced the standard conjectures, the conven- tional wisdom became that, to prove the Riemann hy- 4.6. Note that the residue field κ(s) at a closed point s of S is finite, and so the Künneth components of pothesis in dimension > 1, one should prove the stan- 42 the diagonal are almost-algebraic (3.12). Therefore dard conjectures. However, Deligne recognized the the space of almost-algebraic classes on X is a graded intractability of the standard conjectures and looked Q-subalgebra for other approaches. In 1973 he startled the mathematical world by announcing a proof of the Riemann ∗ M r AA (X) = AA (X) hypothesis for all smooth projective varieties over fi- r≥0 nite fields.43 How this came about is best described 44 ∗ in the following conversation. of the Q-algebra of AH (X) of absolute Hodge classes. For any regular map f : Y → X of complete smooth Gowers: Another question I had. Given the clearly ∗ varieties, the maps f∗ and f send almost-algebraic absolutely remarkable nature of your proof of the last classes to almost-algebraic classes. Similar state- remaining Weil conjecture, it does make one very cu- ments hold for the -algebra of classes almost- Q rious to know what gave you the idea that you had algebraic at p. a chance of proving it at all. Given that the proof was very unexpected, it’s hard to understand how you Beyond Pure Motives could have known that it was worth working on.

In the above I considered only pure motives with Deligne: That’s a nice story. In part because of Serre, rational coefficients. This theory should be general- and also from listening to lectures of Godement, ized in (at least) three different directions. Let k be a I had some interest in automorphic forms. Serre un- 11/2 field. derstood that the p in the Ramanujan conjecture should have a relation with the Weil conjecture itself. • There should be a category of mixed motives A lot of work had been done by Eichler and Shimura, over k. This should be an abelian category whose and by Verdier, and so I understood the connection whose semisimple objects form the category of between the two. Then I read about some work of pure motives over k. Every mixed motive should Rankin, which proved, not the estimate one wanted, be equipped with a weight filtration whose quo- but something which was a 1/4 off—the easy results tients are pure motives. Every algebraic variety were 1/2 off from what one wanted to have. As soon over k (not necessarily nonsingular or projective) as I saw something like that I knew one had to un- should define a mixed motive. derstand what he was doing to see if one could do • There should be a category of complexes of mo- something similar in other situations. And so I looked tives over k. This should be a triangulated cate- at Rankin, and there I saw that he was using a result gory with a t-structure whose heart is the cate- 40 Serre writes (email July 2015): gory of mixed motives. Each of the standard Weil cohomology theories lifts in a natural way to a He was not the first one. So did Weil, around 1965; he functor from all algebraic varieties over k to a tri- looked at surfaces, made some computations, and tried (vainly) to prove a positivity result. He talked to me about angulated category; these functors should factor it, but he did not publish anything. through the category of complexes of motives. 41 • Everything should work mutatis mutandis over Z. “I have no comments on your attempts to generalize the Weil-Castelnuovo inequality…; as you know, I have a Much of this was envisaged by Grothendieck.39 There sketch of a proof of the Weil conjectures starting from the has been much progress on these questions, which curves case.” (Grothendieck, Grothendieck-Serre Correspon- dence, p. 88, 1959). Grothendieck hoped to prove the Weil I shall not attempt to summarize (see André 2004, conjectures by showing that every variety is birationally Mazza et al. 2006, Murre et al. 2013). a quotient of a product of curves, but Serre constructed a counterexample. See Grothendieck-Serre Correspondence, pp. 145–148, 1964, for a discussion of this and Grothen- 5. Deligne’s Proof of the Riemann dieck’s “second attack” on the Weil conjectures. 42 “Pour aller plus loin et généraliser complètement les ré- Hypothesis over Finite Fields sultats obtenus par Weil pour le cas n = 1, il faudrait prouver les propriétés suivantes de la cohomologie `-adique …[the Grothendieck attempted to deduce the Riemann standard conjectures]” Dieudonné 1974, IX n◦19, p. 224. hypothesis in arbitrary dimensions from the curves 43 A little earlier, he had proved the Riemann hypothesis for varieties whose motive, roughly speaking, lies in the cate- 39 Certainly, Grothendieck envisaged mixed motives and mo- gory generated by abelian varieties (Deligne 1972). 44 tives with coefficients in Z (see the footnote on p. 5 of This is my transcription (slightly edited) of part of a tele- Kleiman 1994 and Grothendieck’s letters to Serre and Il- phone conversation which took place at the ceremony an- lusie). nouncing the award of the 2013 Abel Prize to Deligne.

DECEMBER 2016 NOTICES OF THE ICCM 41 of Landau—the idea was that when you had a product is such function of weight 12. Let defining a zeta function you could get information on ∞ the local factors out of information on the pole of the −s Φ f (s) = ∑ c(n)n zeta function itself. The poles were given in various n=1 cases quite easily by Grothendieck’s theory. So then be the Dirichlet series associated with f . Because f is it was quite natural to see what one could do with an eigenfunction this method of Rankin and Landau using that we had information on the pole. I did not know at first how 1 (s) = . far I could go. The first case I could handle was a hy- Φ f ∏ −s 2k−1−2s p prime 1 − c(p)p + p persurface of odd dimension in projective space. But that was a completely new case already, so then I had Write confidence that one could go all the way. It was just 2k−1 2 0 a matter of technique. (1 − c(p)T + p T = (1 − apT)(1 − apT). Gowers: It is always nice to hear that kind of thing. The Ramanujan-Petersson conjecture says that ap and Certainly, that conveys the idea that there was a cer- 0 0 2k−1 ap are complex conjugate. As apap = p , this says tain natural sequence of steps that eventually led to that this amazing proof. k−1/2 ap = p ; Deligne: Yes, but in order to be able to see those steps it was crucial that I was not only following equivalently, lectures in algebraic geometry but some things that |c(n)| ≤ nk−1/2σ (n). looked quite different (it would be less different now) 0 the theory of automorphic forms. It was the discrep- For f = ∆, this becomes Ramanujan’s original conjec- 11 ancy in what one could do in the two areas that gave ture, |τ(p)| ≤ 2p 2 . Hecke showed that, for a cusp form the solution to what had to be done. f of weight 2k, Gowers: Was that just a piece of good luck that you |c(n)| = O(nk). happened to know about both things. There is much to be said about geometric inter- Deligne (emphatically): Yes. pretations of Ramanujan’s function—see Serre 1968 and the letter from Weil to Serre at the end of the Landau, Rankin, and the Ramanujan Conjecture article.

Landau’s Theorem Rankin’s Theorem ∞ −s Let f = c(n)qn be a cusp form of weight 2k, and let Consider a Dirichlet series D(s) = ∑n=1 cnn . Recall ∑ 2 −s that there is a unique real number (possibly ∞ or −∞) F (s) = ∑n≥1 |c(n)| n . Rankin (1939) shows that F (s) such that D(s) converges absolutely for ℜ(s) > σ0 and can be continued to a meromorphic function on the does not converge absolutely for ℜ(s) < σ0. Landau’s whole plane with singularities only at s = k and s = 45 theorem states that, if the cn are real and nonnega- k − 1. Using Landau’s theorem, he deduced that tive, then D(s) has a singularity at s = σ0. For example, k−1/5 suppose that |c(n)| = O(n ).

1 A Remark of Langlands D(s) = c n−s = ; ∑ n ∏ (1 − a p−s)···(1 − a p−s) n≥1 p prime p,1 p,m Langlands remarked (1970, pp. 21–22) that −s Rankin’s idea could be used to prove a generalized if the cn are real and nonnegative, and ∑cnn con- Ramanujan conjecture provided one knew enough verges absolutely for ℜ(s) > σ0, then about the poles of a certain family of Dirichlet se- σ 46 |ap, j| ≤ p 0 , all p, j. ries. 45 Selberg did something similar about the same time. Their Ramanujan’s Conjecture approach to proving the analytic continuation of convolu- n tion L-functions has become known as the Rankin-Selberg Let f = ∑n≥1 c(n)q , c(1) = 1, be a cusp form of method. weight 2k which is a normalized eigenfunction of 46 “The remark about the consequences of functoriality for the Hecke operators T(n). For example, Ramanujan’s Frobenius-Hecke conjugacy classes of an automorphic rep- function resentation, namely that their eigenvalues are frequently all of absolute value of 1, occurred to me on a train platform in ∞ ∞ ∆(τ) = τ(n)qn def= q (1 − qn)24 Philadelphia, as I thought about the famous Selberg-Rankin ∑ ∏ estimate.” Langlands 2015, p. 200. n=1 n=1

42 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 N Given an automorphic representation π = πp Now let U be a connected nonsingular algebraic of an algebraic group G and a representation σ variety over a field k. In this case, there is a notion of of the corresponding L-group, Langlands defines an a local system of Q`-vector spaces on U (often called a L-function smooth or lisse sheaf). For a smooth projective map of algebraic varieties f : V →U and integer r, the direct L s r ( ,σ,π) image sheaf R f∗Q` on U is a locally constant sheaf of ( )m(λ) -vector spaces with fibre 1 Q` = ∏ (power of π) · (Γ−factor) · ∏ −s 1 − λ(tp)p r r λ p prime (R f∗Q`)u ' H (Vu,Q`)

Here λ runs over certain characters, m(λ) is the multi- for all u ∈ U. plicity of λ in σ, and tp is the (Frobenius) conjugacy The étale fundamental group π1(U) of U classifies class attached to . Under certain hypotheses, the 47 πp the finite étale coverings of U. Let E be a local sys- generalized Ramanujan conjecture states that, for all tem on U, and let E = E η be its fibre over the generic λ and p, point η of U. Again, there is a “monodromy” action ρ of π (U) on the finite-dimensional -vector space E, λ(tp) = 1. 1 Q` and the functor E (E,ρ) is an equivalence of cate- Assume that, for all σ, the L-series L(s,σ,π) is an- gories. def alytic for ℜ(s) > 1. The same is then true of D(s,σ) = Now let U0 be a nonsingular geometrically- n om(λ) 1 connected variety over finite field k0; let k be an al- ∏λ ∏p prime 1−λ(t )p−s because the Γ-function has p gebraic closure of k , and let U = (U ) . Then there is no zeros. Let σ = ρ ⊗ ρ¯. The logarithm of D(s,σ) is 0 0 k n ∞ trace σ (tp) −ns an exact sequence ∑p ∑n=1 n p . As

n n n n 2 0 → π1(U) → π1(U0) → Gal(k/k0) → 0. trace σ (tp) = trace ρ (tp) · trace ρ¯ (tp) = |trace ρ (tp)| , The maps reflect the fact that a finite extension k0/k the series for logD(s,σ) has positive coefficients. The 0 pulls back to a finite covering U0 → U of U and a same is therefore true of the series for D(s,σ). We can 0 0 finite étale covering of U0 pulls back to a finite étale now apply Landau’s theorem to deduce that |λ(tp)| ≤ covering of U. p. If λ occurs in some σ, we can choose ρ so that mλ Let be a local system of -vector spaces on U . occurs in . Then (m )(t ) = (t )m is an eigenvalue of E Q` 0 ρ λ p λ p 48 m 2m Grothendieck (1964) proved the following trace for- ρ(tp) and λ(tp) is an eigenvalue of ρ¯; hence λ(tp) 1/2m mula is an eigenvalue of σ, and |λ(tp)| ≤ p for all m. On letting m → ∞, we see that |λ(tp)| ≤ 1 for all λ. Replacing r r (32) ∑ Tr(πu | E u) = ∑(−1) Tr(π | Hc (U,E )). λ with −λ, we deduce that |λ(tp)| = 1. u∈|U|π r Note that it is the flexibility of Langlands’s construction that makes this kind of argument possi- Here |U| denotes the set of closed points of U, πu de- ble. According to Katz (1976, p. 288), Deligne studied notes the local Frobenius element acting on the fibre Rankin’s original paper in an effort to understand this E u of E at u, and π denotes the reciprocal of the usual remark of Langlands. Frobenius element of Gal(k/k0). Let

1 Z(U , ,T) = ; 0 E 0 ∏ deg(u) Grothendieck’s Theorem det(1 − πuT | E u) u∈|U0| Let U be a connected topological space. Recall written multiplicatively, (32) becomes that a local system of Q-vector spaces on U is a sheaf E on U that is locally isomorphic to the constant sheaf r (−1)r+1 n (33) Z(U0,E 0,T) = ∏det(1 − πT | Hc (U,E)) Q for some n. For example, let f : V → U be a smooth r projective map of algebraic varieties over C; for r ≥ 0, r 47 R f∗Q is a locally constant sheaves of Q-vector spaces When the base field is C, the topological fundamental an an with fibre group of V classifies the covering spaces of V (not necessarily finite). The Riemann existence theorem implies that r r the two groups have the same finite quotients, and so the (R f∗Q)u ' H (Vu,Q) étale fundamental group is the profinite completion of the for all u ∈ U( ). Fix a point o ∈ U. There is a canon- topological fundamental group. C 48 ical (monodromy) action ρ of π (U,o) on the finite- In fact, Grothendieck only sketched the proof of his the- 1 orem in this Bourbaki talk, but there are detailed proofs in dimensional Q-vector space E o, and the functor E the literature, for example, in the author’s book on étale co- (E o,ρ) is an equivalence of categories. homology.

DECEMBER 2016 NOTICES OF THE ICCM 43 det(1 − πT | H1(U,E)) (cf. (25), p. 29). For the constant sheaf Q`,(33) be- = c . comes ((28), p. 32). Grothendieck proves (32) by a det(1 − πT | Eπ1(U)(−1)) dévissage to the case that U is of dimension 1.49 There is a canonical pairing of sheaves We shall need (33) in the case that U0 is an affine curve, for example, 1. In this case it becomes A Rd f × Rd f → R2d f ' (−d) (34) ∗Q` ∗Q` ∗Q` Q` 1 det(1 − πT | H1(U,E)) = c . which, on each fibre becomes cup product ∏ deg(u) det(1 − πuT | E u) det(1 − πT | E (U)(−1)) u∈|U0| π1 d d 2d H (V u,Q`) × H (V u,Q`) → H (V u,Q`) ' Q`(−d). Here Eπ1(U) is the largest quotient of E on which π1(U) acts trivially (so the action of π1(U0) on it factors This gives a pairing through Gal(k/k0)). ψ : E0 × E0 → Q`(−d)

Proof of the Riemann Hypothesis for which is skew-symmetric (because d is odd), non- hypersurfaces of Odd Degree degenerate (by Poincaré duality for the geometric Following Deligne, we first consider a hypersur- generic fibre of f ), and π1(U0)-invariant (because it d+1 arises from a map of sheaves on U ). Deligne proves face V0 in Pk of odd degree δ. For a hypersurface, the 0 0 N cohomology groups coincide with those of the ambi- that, if the line through P0 in P is chosen to be suffi- 50 ent space except in the middle degree, and so ciently general, then the image of π1(U) in Sp(E,ψ) is dense for the Zariski topology (i.e., there is “big ge- P (V,T) d ometric monodromy”). Thus the coinvariants of π1(U) Z(V,T) = d , (1 − T)(1 − qT)···(1 − q T) in E are equal to the coinvariants of Sp(E,ψ), and the invariant theory of the symplectic group is well un- with P (V,T) = det(1 − πT | Hd(V, )). Note that d Q` derstood. d Note that Pd(V,T) = Z(V,T)(1 − T)···(1 − q T) ∈ Q[T]. deg u n n (T ) We embed our hypersurface in a one-dimensional logZ(U0,E 0,T) = ∑ ∑Tr(πu | Eu) . n n family. The homogeneous polynomials of degree δ in u∈|U0| d + 2 variables, considered up to multiplication by a n N The coefficients Tr(πu | Eu) are rational. When we re- nonzero scalar, form a projective space P with N = ⊗2m n place E with E , we replace Tr(π | E ) with d+δ+1 . There is a diagram 0 0 u u δ n ⊗2m n 2m Tr(πu | Eu ) = Tr(πu | Eu) N d+ H P × P 1 f which is ≥ 0. To apply Landau’s theorem, we need to find the N ⊗2m −s P poles of Z(U0,E0 ,q ), which, according to Grothen- dieck’s theorem (see (35)) are equal to the zeros of N in which the fibre HP over a point P of P (k) is the hy- det(1 − πT | E (−1)). Note that d+ π1(U) persurface (possibly reducible) in P 1 defined by P. Let P0 be the polynomial defining V0. We choose a ⊗2m Sp(ψ) ⊗2m Hom(E ,Q`) = Hom((E )Sp(ψ),Q`). N “general” line through P0 in P and discard the finitely many points where the hypersurface is singular of not The map connected. In this way, we obtain a smooth projective 1 ⊗2m map f : V 0 → U0 ⊂ P whose fibres are hypersurfaces x1 ⊗···⊗x2m 7→ ψ(x1,x2)···ψ(x2m−1,x2m): E → Q`(−dm) d+1 of degree δ in ; let o ∈ U0 be such that the fibre P ⊗2m Sp(ψ) is invariant under Sp(ψ), and Hom(E , `) has a over o is our given hypersurface V0. Q d basis of maps { f1,..., fM}obtained from this one by Let E 0 = R f∗Q` and let E denote the correspond- permutations. The map ing π1(U0)-module. Then ⊗2m M 1 (36) a 7→ ( f1(a),..., fM(a)): E → Q`(−dm) (35) Z(U , ,T) = 0 E 0 ∏ deg(u) u∈|U | Pd(Vu,T ) 0 induces an isomorphism 49 This proof of the rationality of the zeta function of an arbi- ⊗2m M trary variety is typical of Grothendieck: find a generalization (E )Sp(ψ) → Q`(−dm) . that allows the statement to be proved by a “devissage” to 50 the (relative) dimension one case. This may require a finite extension of the base field k0.

44 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 0 The Frobenius element π ∈ Gal(k/k0) acts on Q`(−md) for all complex conjugates α of α. Then Theorem 5.1 md ⊗2m md+1 as q , and so it acts on E (−1) as q . We can holds for all nonsingular projective varieties over q. π1(U0) F now apply Landau’s theorem to deduce that Proof. Let V0 be a smooth projective variety of dimen- α2m ≤ qmd+1 sion d (not necessarily even) over Fq, and let α be an d eigenvalue of π on H (V,Q`). The Künneth formula d m for every eigenvalue α of πo acting on E o = H (V0,Q`). shows that α occurs among the eigenvalues of π act- dm m On taking the 2mth root and letting m → ∞, we find ing on H (V ,Q`) for all m ∈ N. The hypothesis in the d/2 d that |α| ≤ q . As q /α is also an eigenvalue of πo on proposition applied to an even m shows that d d/2 H (V0,Q`), we deduce that |α| = q . md − 1 0 m md + 1 q 2 2 < |α | < q 2 2

Proof of the Riemann Hypothesis for Nonsingular for all conjugates α0 of α. On taking the mth root and Projective Varieties letting m tend to infinity over the even integers, we We shall prove the following statement by induc- find that tion on the dimension of V . According to the discus- 0 d 0 |α | = q 2 . sion in (2.7), it completes the proof of Conjectures r W1–W4. Now let α be an eigenvalue of π acting on H (V,Q`). If r > d, then αd is an eigenvalue of π acting on Theorem 5.1. Let V0 be an nonsingular projective variety over . Then the eigenvalues of π acting on r ⊗d 0 ⊗r−d rd r Fq H (V, `) ⊗ H (V, `) ⊂ H (V , `) r Q Q Q H (V,Q`) are algebraic numbers, all of whose complex r/2 0 d conjugates have absolute value q . because π acts as 1 on H (V,Q`). Therefore α is alge- d rd The Main Lemma (Restricted Form) braic and |α | = q 2 , and similarly for its conjugates. The case r < d can be treated similarly using that π The following is abstracted from the proof of Rie- d 2d acts as q on H (V,Q`), or by using Poincaré duality.2 mann hypothesis for hypersurfaces of odd degree. Completion of the Proof Main Lemma 5.2. Let E0 be a local system of Q`-vector spaces on U0, and let E be the corresponding To deduce Theorem 5.1 from (5.2), Deligne uses π1(U0)-module. Let d be an integer. Assume: the theory of Lefschetz pencils and their cohomology (specifically vanishing cycle theory and the Picard- (i) (Rationality.) For all closed points u ∈ U , the char- 0 Lefschetz formula). In the complex setting, this the- acteristic polynomial of π acting on has ratio- u E u ory was introduced by Picard for surfaces and by nal coefficients. Lefschetz (1924) for higher dimensional varieties. It (ii) There exists a nondegenerate π (U )-invariant 1 0 was transferred to the abstract setting by Deligne and skew-symmetric form Katz in SGA 7 (1967–1969). Deligne had earlier used these techniques to prove the following weaker re- ψ : E × E → Q`(−d). sult: (iii) (Big geometric monodromy.) The image of π1(U) let V0 be a smooth projective variety over a finite field in Sp(E,ψ) is open. k0 of odd characeristic that lifts, together with a polarization, to characteristic zero. Then the polynomials det(1−πT | r Then: H (V,Q`)) have integer coefficients independent of ` (Verdier 1972). (a) For all closed points u ∈ U0 , the eigenvalues of πu deg u d/2 acting on Eu have absolute value (q ) . Thus, Deligne was already an expert in the application (b) The characteristic polynomial of π acting on of these methods to zeta functions. 1 Hc (U,E ) is rational, and its eigenvalues all have Let V be a nonsingular projective variety V of di- absolute value ≤ qd/2+1. mension d ≥ 2, and embed V into a projective space m m ⊗2m P . The hyperplanes H : ∑aiTi = 0 in P form a projec- Proof. As before, E is isomorphic to a direct sum m π1(U) tive space, called the dual projective space Pˇ .A pencil of copies of Q`(−md), from which (a) follows. Now (b) 1 of hyperplanes is a line D = {αH0 +βH∞ | (α : β) ∈ P (k)} follows from (34). 2 m in Pˇ . The axis of the pencil is A Reduction \ A = H0 ∩ H∞ = Ht . Proposition 5.3. Assume that for all nonsingular va- t∈D rieties V0 of even dimension over Fq, every eigenvalue dimV Such a pencil is said to be a Lefschetz pencil for V if α of π on H (V,Q`) is an algebraic number such that (a) the axis A of the pencil cuts V transversally; (b) the dim V − 1 0 dim V + 1 def q 2 2 < |α | < q 2 2 hyperplane sections Vt = V ∩Dt are nonsingular for all

DECEMBER 2016 NOTICES OF THE ICCM 45 t in some open dense subset U of D; (c) for all t ∈/ U, 6. The Hasse-Weil Zeta Function the hyperplane section V has only a single singular- t The Hasse-Weil Conjecture ity and that singularity is an ordinary double point. Given such a Lefschetz pencil, we can blow V up along Let V be a nonsingular projective variety over a the A ∩V to obtain a proper flat map number field K. For almost all prime ideals p in OK (called good), V defines a nonsingular projective vari- ∗ f : V → D ety V(p) over the residue field κ(p) = OK/p. Define the zeta function of V to be such that the fibre over t in D is Vt = V ∩ Ht . We now prove Theorem 5.1 (and hence the Rie- ζ(V,s) = ∏ ζ(V(p),s). mann hypothesis). Let V0 be a nonsingular projective p good variety of even dimension d ≥ 2 over a finite field k0. For example, if V is a point over Q, then ζ(X,s) is Embed V0 in a projective space. After possibly replac- 1 the original Riemann zeta function ∏ −s . Using ing the embedding with its square and k with a finite p 1−p 0 the Riemann hypothesis for the varieties V(p), one extension, we may suppose that there exists a Lef- sees that the that the product converges for ℜ(s) > schetz pencil and hence a proper map f : V ∗ → D = 1. 0 P dim(V)+1. It is possible to define factors correspond- Recall (5.3) that we have to prove that ing to the bad primes and the infinite primes. The qd/2−1/2 < |α| < qd/2+1/2 completed zeta function is then conjectured to extend analytically to a meromorphic function on the d for all eigenvalues α on H (V,Q`). It suffices to do this whole complex plane and satisfy a functional equa- ∗ tion. The function (V,s) is usually called the Hasse- with V0 replaced by V0 . From the Leray spectral se- ζ quence, Weil zeta function of V, and the conjecture the Hasse- Weil conjecture. Hr( 1,Rs f ) =⇒ Hr+s(V, ), P ∗Q` Q` More precisely, let r be a natural number. Serre we see that it suffices to prove a similar statement for (1970) defined: each of the groups (a) polynomials Pr(V(p),T) ∈ Z[T] for each good prime 2 1 d−2 1 1 d−1 0 1 d (namely, those occurring in the zeta function for H (P ,R f∗Q`), H (P ,R f∗Q`), H (P ,R f∗Q`). V(p)); The most difficult group is the middle one. Here (b) polynomials Qp(T) ∈ Z[T] for each bad prime p; Deligne applies the Main Lemma to a certain quotient (c) gamma factors Γv for each infinite prime v (de- d−1 pending on the Hodge numbers of V ⊗ k ); subquotient E of R f∗Q`. The hypothesis (i) of the k v Main Lemma is true from the induction hypothesis; (d) a rational number A > 0. the skew-symmetric form required for (ii) is provided Set by an intersection form; finally, for (iii), Deligne was 1 able to appeal to a theorem of Kazhdan and Margulis. ξ(s) = As/2 · ∏ P (V(p),Np−1) Now the Main Lemma shows that |α| ≤ qd/2+1/2 for an p good r eigenvalue of π on H1(P1,E ), and a duality argument 1 d−1/2 · ∏ −1 · ∏ Γv(s). shows that q ≤ |α|. p bad Qp(Np ) v inﬁnite Notes. Deligne’s proof of the Riemann hypothesis The Hasse-Weil conjecture says that ζ (s) extends to over finite fields is well explained in his original paper r (Deligne 1974) and elsewhere. There is also a purely a meromorphic function on the whole complex plane p-adic proof (Kedlaya 2006). and satisfies a functional equation ξ(s) = wξ(r + 1 − s), w = ±1. Beyond the Riemann Hypothesis Over Finite Fields In a seminar at IHES, Nov. 1973 to Feb. 1974, History Deligne improved his results on zeta functions. In According to Weil’s recollections (Œuvres, II, particular he proved stronger forms of the Main p. 529),51 Hasse defined the Hasse-Weil zeta function Lemma. These results were published in Deligne for an elliptic curve over Q, and set the Hasse-Weil 1980, which has become known as Weil II. Even beyond his earlier results on the Riemann hypothesis, 51 Peu avant la guerre, si mes souvenirs sont exacts, G. de these results have found a vast array of applications, Rham me raconta qu’un de ses étudiants de Genève, Pierre which I shall not attempt to summarize (see the var- Humbert, était allé à Göttingen avec l’intention d’y travailler sous la direction de Hasse, et que celui-ci lui avait proposé un ious writings of Katz, especially Katz 2001, and the probléme sur lequel de Rham désirait mon avis. Une courbe book Kiehl and Weissauer 2001). elliptique C étant donnée sur le corps des rationnels, il s’agis-

46 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 conjecture in this case as a thesis problem! Initially, “Yes!”. Although no general proof is available, this response Weil was sceptical of the conjecture, but he proved to the available examples and partial proofs is fully justi- fied.54 (Langlands 2015, p. 205.) it for curves of the form Y m = aXn + b over number fields by expressing their zeta functions in terms of Hecke L-functions.52 In particular, Weil showed that The Functoriality Principle and Conjectures on the zeta functions of the elliptic curves Y 2 = aX3 + b Algebraic Cycles and Y 2 = aX4 + b can be expressed in terms of Hecke As noted earlier, Langlands attaches an L-func- L-functions, and he suggested that the same should tion L(s,σ,π) to an automorphic representation π of be true for all elliptic curves with complex multipli- a reductive algebraic group G over Q and a represen- cation. This was proved by Deuring in a “beautiful se- tation σ of the corresponding L-group LG. The func- ries” of papers. toriality principle asserts that a homomorphism of Deuring’s result was extended to all abelian va- L-groups LH → LG entails a strong relationship be- rieties with complex multiplication by Shimura and tween the automorphic representations of H and G. Taniyama and Weil. This remarkable conjecture includes the Artin conjec- In a different direction, Eichler and Shimura ture, the existence of nonabelian class field theories, proved the Hasse-Weil conjecture for elliptic modu- and many other fundamental statements as special lar curves by identifying their zeta functions with the cases. And perhaps even more. Mellin transforms of modular forms. Langlands doesn’t share the pessimism noted Wiles et al. proved Hasse’s “thesis problem” as earlier concerning the standard conjectures and the part of their work on Fermat’s Last Theorem (for a conjectures of Hodge and Tate. He has suggested that popular account of this work, see Darmon 1999). the outstanding conjectures in the Langlands pro- gram are more closely entwined with the outstanding Automorphic L-Functions conjectures on algebraic cycles than is usually recog- The only hope one has of proving the Hasse-Weil nized, and that a proof of the first may lead to a proof conjecture for a variety is by identifying its zeta func- of the second. tion with a known function, which has (or is conjec- I believe that it is necessary first to prove functoriality tured to have) a meromorphic continuation and func- and then afterwards, with the help of the knowledge and tools obtained, to develop a theory of correspondences and tional equation. In a letter to Weil in 1967, Langlands simultaneously of motives over Q and other global fields, as 55 defined a vast collection of L-functions, now called well as C. (Langlands 2015, p. 205). automorphic (Langlands 1970). He conjectures that In support of this statement, he mentions two ex- all L-functions arising from algebraic varieties over amples (Langlands 2007, 2015). The first, Deligne’s number fields (i.e., all motivic L-functions) can be ex- proof of the Riemann hypothesis over finite fields, pressed as products of automorphic L-functions.53 has already been discussed in some detail. I now dis- The above examples and results on the zeta functions cuss the second example. of Shimura varieties have made Langlands very optimistic: The Problem With the help of Shimura varieties mathematicians cer- Let A be a polarized abelian variety with complex tainly have, for me, answered one main question: is it pos- al sible to express all motivic L-functions as products of auto- multiplication over Q . The theory of Shimura and morphic L-functions? The answer is now beyond any doubt, Taniyama describes the action on A and its torsion points of the Galois group of al over the reflex field sait principalement, il me semble, d’étudier le produit infini Q of A. In the case that A is an elliptic curve, the reflex des fonctions zêta des courbes Cp obtenues en réduisant C modulo p pour tout nombre premier p pour lequel Cp est de field is a quadratic imaginary field; since one knows genre 1; plus précisément, il fallait rechercher si ce produit how complex conjugation acts, in this case we have possède un prolongement analytique et une équation fonc- tionnelle. J’ignore si Pierre Humbert, ou bien Hasse, avaient 54 Mit Hilfe der Shimuravarietäten haben die Mathematiker examiné aucun cas particulier. En tout cas, d’après de Rham, gewiß eine, für mich, Hauptfrage beantwortet: wird es Pierre Humbert se sentait découragé et craignait de perdre möglich sein, alle motivischen L-Funktionen als Produkte son temps et sa peine. von automorphen L-Funktionen auszudrücken? Die Antwort 52 Weil also saw that the analogous conjecture over global ist jetzt zweifellos, “Ja!”. Obwohl kein allgemeiner Beweis function fields can sometimes be deduced from the Weil con- vorhanden ist, ist diese Antwort von den vorhanden Beispie- jectures. len und Belegstücken her völlig berechtigt. 53 Or perhaps there should even be an identification of the 55 Ich habe schon an verschiedenen Stellen behauptet, daßes tannakian category defined by motives with a subcategory of meines Erachtens nötig ist, erst die Funktorialität zu be- a category defined by automorphic representations. (“Wenn weisen und dann nachher, mit Hilfe der damit erschlosse- man sich die Langlands-Korrespondenz als eine Identifika- nen Kenntnisse und zur Verfügung gestellten Hilfsmittel, tion einer durch Motive definierten Tannaka-Kategorie mit eine Theorie der Korrespondenz und, gleichzeitig, der Mo- einer Unterkategorie einer durch automorphe Darstellungen tive, über Q und anderen globalen Körper, sowie über C zu definierten Kategorie vorstellt…” Langlands 2015, p. 201.) begründen.

DECEMBER 2016 NOTICES OF THE ICCM 47 a description of how the full Galois group Gal(Qal/Q) the one-cocycle also gives a construction of a conjec- acts. Is it possible to find such a description in higher tural Taniyama group. The descriptions of the action dimensions? of Gal(Qal/Q) on CM abelian varieties and their tor- Shimura studied this question, but concluded sion points given by the motivic Taniyama group and rather pessimistically that “In the higher-dimensional by Langlands’s conjectural Taniyama group are both case, however, no such general answer seems possi- consistent with the results of Shimura and Taniyama. ble.” However, Grothendieck’s theory of motives sug- Deligne proved that this property characterizes the gests the framework for an answer. The Hodge con- groups, and so the two are equal. This solves the jecture implies the existence of tannakian category of problem posed above. CM-motives over Q, whose motivic Galois group is an Concerning Langlands’s conjugacy conjecture it- extension self, this was proved in the following way. For those Shimura varieties with the property that each con- al 1 → S → T → Gal(Q /Q) → 1 nected component can be described by the moduli of abelian varieties, Shimura’s conjecture was proved of Gal(Qal/Q) (regarded as a pro-constant group in many cases by Shimura and his students and in scheme) by the Serre group S (a certain pro-torus); general by Deligne. To obtain a proof for a general étale cohomology defines a section to T → Gal(Qal/Q) Shimura variety, Piatetski-Shapiro suggested embed- over the finite adèles. Let us call this entire system ding the Shimura variety in a larger Shimura vari- the motivic Taniyama group. This system describes ety that contains many Shimura subvarieties of type al how Gal(Q /Q) acts on CM abelian varieties over Q A1. After Borovoi had unsuccessfully tried to use and their torsion points, and so the problem now be- Piatetski-Shapiro’s idea to prove Shimura’s conjec- comes that of giving an explicit description of the mo- ture directly, the author used it to prove Langlands’s tivic Taniyama group. conjugation conjecture, which has Shimura’s conjecture as a consequence. No direct proof of Shimura’s The Solution conjecture is known. Shimura varieties play an important role in the work of Langlands, both as a test of his ideas and Epilogue for their applications. For example, Langlands writes (2007): Stepanov (1969) introduced a new elementary approach for proving the Riemann hypothesis for Endoscopy, a feature of nonabelian harmonic analysis on reductive groups over local or global fields, arose implicitly curves, which requires only the Riemann-Roch the- in a number of contexts… It arose for me in the context of orem for curves. This approach was completed and the trace formula and Shimura varieties. simplified by Bombieri (1974) (see also Schmidt Langlands formulated a number of conjectures con- 1973). cerning the zeta functions of Shimura varieties. How would our story have been changed if this One problem that arose in his work is the fol- proof had been found in the 1930s? lowing. Let S(G,X) denote the Shimura variety over C defined by a Shimura datum (G,X). According to Acknowledgement the Shimura conjecture S(G,X) has a canonical model I thank F. Oort and J-P. Serre for their comments S(G,X)E over a certain algebraic subfield E of C. The on the first draft of this article, and Serre for con- Γ-factor of the zeta function of S(G,X)E at a com- tributing the letter of Weil. It would not have been plex prime v: E ,→ C depends on the Hodge the- possible to write this article without the University of ory of the complex variety v(S(G,X)E ). For the given Michigan Science Library, which provided me with a embedding of E in C, there is no problem because copy of the 1953 thesis of Frank Quigley, and where v(S(G,X)E ) = S(G,X), but for a different embedding, I was able to find such works as Castelnuovo’s Mem- what is v(S(G,X)E )? This question leads to the following problem: oire Scelte … and Severi’s Trattato ... available on open shelves. Let σ be an automorphism of C (as an abstract field); how can we realize σS(G,X) as the Shimura variety attached to another (explicitly defined) Shimura datum (G0,X0)? References In his Corvallis talk (1979), Langlands states a “Colmez and Serre 2001” is cited as “Grothendieck-Serre Cor- conjectural solution to this problem. In particular, respondence”. he constructs a “one-cocycle” which explains how to Abdulali, S. 1994. Algebraic cycles in families of abelian varieties. Canad. J. Math. 46, No. 6, 1121–1134. 0 0 twist (G,X) to obtain (G ,X ). When he explained his André, Y. 1996. Pour une théorie inconditionnelle des motifs. construction to Deligne, the latter recognized that Inst. Hautes Études Sci. Publ. Math. No. 83, 5–49.

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DECEMBER 2016 NOTICES OF THE ICCM 49 Katz, N. 2001. L-functions and monodromy: four lectures on Quigley, F. D. 1953. On Schubert’s Formula, thesis, University Weil II. Adv. Math. 160, No. 1, 81–132. of Chicago. Kedlaya, K. 2006. Fourier transforms and p-adic ‘Weil II’. Rankin, R. 1939. Contributions to the theory of Ramanujan’s Compos. Math. 142, No. 6, 1426–1450. function τ(n) and similar arithmetical functions. I. The ze- ∞ s 13 Kiehl, R.; Weissauer, R, 2001. Weil conjectures, perverse ros of the function ∑n=1 τ(n)/n on the line ℜs = 2 . II. The sheaves and l’adic (sic) Fourier transform. Springer-Verlag, order of the Fourier coefficients of integral modular forms. Berlin. Proc. Cambridge Philos. Soc. 35, 351–372. Kleiman, S. 1968. Algebraic cycles and the Weil conjectures. Roquette, P. 1953. Arithmetischer Beweis der Riemannschen Dix exposés sur la cohomologie des schémas, pp. 359–386, Vermutung in Kongruenzfunktionenkörpern beliebigen North-Holland, Amsterdam; Masson, Paris. 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50 NOTICES OF THE ICCM VOLUME 4,NUMBER 2 Tate, J. 1964. Algebraic cohomology classes, 25 pages. In: Finally, I append a letter from Weil to Serre con- Lecture notes prepared in connection with seminars held cerning the “variété de Ramanujan”. At the request of at the Summer Institute on Algebraic Geometry, Whitney the editors, I include a translation. Estate, Woods Hole, MA, July 6–July 31, 1964. Tate, J. 1966a. On the conjectures of Birch and Swinnerton- 12th February, 1960. Dyer and a geometric analog. Séminaire Bourbaki, Vol. 9, Exp. No. 306, 415–440. My dear Serre, Tate, J. 1966b. Endomorphisms of abelian varieties over fi- Let V be a (complete, nonsingular) polarized va- nite fields. Invent. Math. 2, 134–144. Tate, J. 1968. Classes d’isogénie des variétés abéliennes sur riety; let f be a morphism from V to V multiplying un corps fini (d’après T. Honda) Séminaire Bourbaki 352, the fundamental class by q = q( f ); let E = E( f ,d) be (1968/69). the endomorphism determined by f on the homology Tate, J. 1994. Conjectures on algebraic cycles in `-adic coho- group of V of topological dimension d; let F(t) = F (t) mology. Motives (Seattle, WA, 1991), 71–83, Proc. Sympos. f ,d Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI. be the characteristic polynomial of E, det(1 − t.E). As Verdier, J-L. 1972. Indépendance par rapport à ` des you skillfully proved, the roots of F(t) have absolute polynômes caractéristiques des endomorphismes de value of qd/2. Frobenius de la cohomologie `-adique (d’après P. Deligne). Suppose that there exists a V, defined over Z, with Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 423, pp. 98–115. the following wondrous property: when we reduce V Voevodsky, V. 1995. A nilpotence theorem for cycles alge- modulo p, and we take d to be 11 and f to be the Frobe- braically equivalent to zero. Internat. Math. Res. Notices nius morphism of V reduced mod.p, then 1995, No. 4, 187–198. Weil, A. 1940. Sur les fonctions algébriques à corps de con- F(t) = 1 − τ(p).t + p11t2 stantes fini. C. R. Acad. Sci. Paris 210, 592–594. Weil, A. 1941. On the Riemann hypothesis in function-fields. Proc. Nat. Acad. Sci. U. S. A. 27, 345–347. with τ(p) the Ramanujan function. It is known that Weil, A. 1946. Foundations of Algebraic Geometry. Ameri- this satisfies remarkable congruences, which I shall can Mathematical Society Colloquium Publications, Vol. 29. write in my way. Put, for a given p and arbitrary i, American Mathematical Society, New York. q = pi, f = f i (ith iterate of the Frobenius, i.e., the Weil, A. 1948a. Sur les courbes algébriques et les variétés qui q s’en déduisent. Actualités Sci. Ind., No. 1041 = Publ. Inst. Frobenius over the field with q elements), Fq = Ffq,11(t). 8 Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris. Then we have: Fq(1) ≡ 0 (691); Fq(1) ≡ 0 (2 ) if p 6= 2; Weil, A. 1948b. Variétés abéliennes et courbes algébriques. Fq(1) ≡ 0 (23) if (q/23) = −1 (quadratic residue symbol; Actualités Sci. Ind., No. 1064 = Publ. Inst. Math. Univ. Stras- I find no indication in the literature for other values bourg 8 (1946). Hermann & Cie., Paris. −1 2 2 −2 2 Weil, A. 1948c. On some exponential sums. Proc. Nat. Acad. of q); q.Fq(q ) ≡ 0 (5 .7); q .Fq(q ) ≡ 0 (3 ). As far as I Sci. U. S. A. 34, 204–207. know, this is the complete list of known congruences. Weil, A. 1949. Numbers of solutions of equations in finite Questions: (a) can one prove, in the classical case, fields. Bull. Amer. Math. Soc. 55, 497–508. congruences for F(t) having the general shape of the Weil, A. 1950. Variétés abéliennes. Algèbre et théorie des nombres. Colloques Internationaux du Centre National de ones I have just written? (b) can one, starting from la Recherche Scientifique, No. 24, pp. 125–127. Centre Na- there, predict what should be the variety V corre- tional de la Recherche Scientifique, Paris. sponding, in the way described above, to the Ramanu- Weil, A. 1952. Fibre spaces in algebraic geometry (Notes by jan function? A. Wallace). University of Chicago, (mimeographed) 48 pp. Weil, A. 1955. On algebraic groups of transformations. Amer. Note that 691 is the numerator of B6, and that J. Math. 77, 355–391. the product 23.691 appears in Milnor’s formula for Weil, A. 1958. Introduction à l’étude des variétés kähléri- the number of differentiable structures on S23! As ennes. Publications de l’Institut de Mathématique de l’Uni- Hirzebruch told me that 691 appears only beginning versité de Nancago, VI. Actualités Sci. Ind. No. 1267 Her- mann, Paris. in topological dimension 24, this suggests that the Weil, A., Œuvres Scientifiques, Collected papers. Springer- “Ramanujan variety” may have dimension 12 (??? of Verlag, New York-Heidelberg, Corrected second printing course τ corresponds to a modular form of degree 12 1980. (For the first printing (1979), it may be necessary as everyone knows). to subtract 4 from some page numbers.) Zariski, O. 1952. Complete linear systems on normal vari- Best regards, A. W. eties and a generalization of a lemma of Enriques-Severi. Ann. of Math. (2) 55, 552–592.

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