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Frans Oort The Weil NAW 5/15 nr. 3 september 2014 211

Frans Oort Mathematisch Instituut Universiteit Utrecht [email protected]

Geschiedenis The

As of 2003, the Abel Prize has been awarded yearly to one or more mathematicians. Up I expect that the first four pages can be un- to 2014, fourteen laureates have received this prize, which has often been described as the derstood by a general audience; the next two mathematician’s ‘Nobel prize’. In 2013 it was awarded to . This event was pages will convey some of the thrill of this celebrated by a presentation during a WONDER meeting in Delft, in December 2013. The present daring by Weil; the last two pages note of Frans Oort aims to give a part of that presentation, in particular a contemplation on the give ample references about recent work on ‘flow of ’ which led to this great success: the main topic will be on the ‘pre-history’ the Weil conjectures. Excellent surveys are of the Weil conjectures. [20, 26, 33, 40], also see [8].

The fundamental problem in number theo- Below, we will discuss the history of this flow Counting points on varieties over finite fields ry is surely how to solve equations in inte- of ideas. This was considered by Gauss, DA 357 (Dis- gers. Since this question is still largely inac- quisitiones Arithmeticae), and in his famous cessible, we shall content ourselves with the Remark. Does this provide any progress for ‘Last Entry’ 146 in 1814 in his Tagebuch: the problem of solving polynomial congruences the solution of the classical RH? The answer modulo p. in [20] is negative in the strict sense: there is no im- plication pRH RH (and also no argument the ⇒ The (RH) has been in reverse way). However, it may give us confi- the focus of mathematical research ever since dence of being on the right track. Moreover, Bernhard Riemann stated this conjecture in tools have been developed, several techni- 1859. However, this problem will not be con- cal steps have been made, and above all the sidered here. From 1924 onward an analogue deep insight in arithmetic aspects of geom- of the RH has been studied, usually called the etry have proved to be a powerful aspect of characteristic p formulation. In order to avoid modern mathematics. any confusion we will refer to this by pRH (al- We discuss: though this is not standard). Basically, this Formulation of the (equivalent of the) RH − problem concerns solving equations over fi- by , and results by F.K. Schmidt, nite fields. We will see that the pRH answers Hasse and Weil proving this conjecture the question of counting the number of so- in special cases (algebraic curves and lutions to polynomial equations over all finite abelian varieties defined over finite fields). fields of the same characteristic at one stroke. We indicate how this motivated Andre´ Weil − This will be illustrated by an easy example, to to formulate his conjectures. be followed throughout the paper. In short sections we give references for re- − After fifty years Pierre Deligne placed the sults proved by Grothendieck with many

crown on this impressive series of develop- of his co-workers, and finally by Deligne, Photo: IHES ments by proving the pRH in its full generality. proving the Weil conjectures. Pierre Deligne during his time at IHES

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number of rational points on several elliptic Euler series and the curves (in our terminology) over a prime field Euler studied in 1740 infinite series defined Fp were computed [11–12]. Also other math- as an infinite sum of positive numbers, and ematicians considered such cases. later Chebyshev studied this as a function of Let me take one simple example, which we a real variable s: will follow throughout our journey. Try to solve

the equation ∞ 1 1 = , s , X ns Y 1 p s ∈ R 2 3 2 n=1 p − Y Y = X X , with x, y 2m . − − − ∈ F

It is clear that over F2 there are exactly four where p runs through the set of all prime num- solutions given by x = 0, 1 and y = 0, 1; bers. Both sides converge for s > 1.

note that 1 = +1 2. We can embed Riemann, who knew work by Euler and who − ∈ F the affine plane into the projective plane by knew the relation of this function with the the- (x, y) = [x:y:1]. We know that, adding the ory of prime numbers, wrote in 1859 his mas- ‘point at infinity’ [0:1:0] (the unique point at terpiece Über die Anzahl der Primzahlen unter ‘infinity’ on the line X = 0) we obtain an ellip- einer gegebenen Grösse [31]. Riemann con- 2 Emil Artin (1898–1962) tic curve D defined over K = 2 by the sidered the function ⊂ PK F homogeneous equation an ideal I Z[T1,...,Tm] such that R = ∞ 1 1 ⊂ ∼ 2 2 3 2 = ζ(s) = = , s , [T1,...,Tm]/I. Here are some examples: , Y Z YZ X X Z. X ns Y 1 p s ∈ C Z Z − − n=1 p − − Z[T ], Fp (where Fp = Z/p), Fp[T ], etc. We write our modest result as convergent for the real part of s greater than Lemma. Let R be an algebra of finite type over #(D( )) = 5. . For any maximal ideal M R the residue F2 one, and he proved that this complex function Z ⊂ has an analytic continuation. He showed that class ring R/M is a finite field. How do we compute Nm := #(D(F2m )) for (the analytic continuation of) this function has For a ring R as above we define its ‘zeta all m >0? We will show that abstract zeros at all negative, even, integral values of ∈ Z theory gives a complete (and easy) answer s (the ‘trivial zeros’). Riemann then stated function’ by to this question (and in fact, the same his famous conjecture that for this function 1 ζ (s) = , method answers this question over any finite (called the zeta function by Riemann) (con- R Y 1 #(R/M) s M − field). tinued across the pole at s = 1) all ‘non-trivial − zeros’ should have real part with absolute val- 1 where this (Euler) product ranges over all max- Explanation. A finite field is a field with a ue equal to 2 (in this case ‘non-trivial zeros’ imal ideals M R, and where s is a complex finite number of elements. An example is the can be understood as zeros with imaginary ⊂ variable. E.g. see [38]. residue class ring Z/p, where p is a prime part not equal to zero). It is impressive to number (and indeed, as is easily proved, this read this paper by Riemann: concise, a mas- Examples. 1. In case R = we have the clas- is a field). However Z/4 is not a field: it has terpiece of technical skill. This conjecture, Z zero-divisors, the element 2 mod 4 does not if true, would give a insight in the distribu- sical Riemann zeta function have an inverse. General theory says that for tion of prime numbers. His idea, his conjec- ζZ(s) = ζ(s): any prime power q = pn, where p is a prime ture, now indicated by RH, has had an enor- number and n Z>0, there exists a finite mous impact on mathematics, and it still has ∈ any maximal ideal M R = is of the form field with q elements, and this field is unique this influence. ⊂ Z M = (p) Z, where p is a prime number, and up to isomorphism; it will be denoted by Fq, ⊂ in particular Fp = Z/p. Remark. These series were considered by Eu- An E over a field K is a non- ler, already in 1740, for positive integral values 1 ζZ(s) = 2 Y s singular E given by a of s. As far as I know Euler did not consider 1 #(R/M)− ⊂ PK M − K s cubic equation having at least one -rational this as a ‘function’ in , therefore I will not 1 ∞ 1 2 = = . point. The curve D P given by the homo- use the terminology ‘the Euler zeta-function’. Y 1 p s X ns ⊂ p − 1 geneous equation ZY 2 Z2Y = X3 ZX2 Dirichlet and Chebyshev studied such series − − − (over a field in which 11 = 0) is an example of as a function (of a complex or a real variable). 6 an elliptic curve. 2. For the ring of integers in a number field we Side remark: This equation considered The Dedekind and E. Artin zeta functions obtain what is now called the Dedekind zeta over F11 defines a curve on which (x = At the moment we have no idea how to prove function. 8 mod 11, y = 6 mod 11) is the (only) sin- (or disprove?) the Riemann Hypothesis for the 3. For a ring like R = Z[T ], or Fp[T ], we obtain gular point (a node). Riemann zeta function. a new type of zeta function. Remarkably enough, it might help to study First, we will follow a different line of de- a generalization. Let R be an algebra of fi- For these zeta functions (e.g. where Z is a velopment in history. nite type over Z. This means that there is subring of R) we hope we can extend classi-

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cal properties of the Riemann zeta function: Here we set up notation. As base field 1 ZV (T ) = . they should extend to a meromorphic func- we choose K = , where q is a power of p. 1 qdT Fq − tion over the complex plane and they should For any m >0 we write Km := qm , i.e. ∈ Z F satisfy a functional equation similar to that K Km is the extension of degree m (unique 1 ⊂ Example. For V = PK , the projective line, we of the classical Riemann zeta function. We up to a K-isomorphism). j have Nj = q + 1 and we see that can ask for their non-trivial zeros (the ex- tended Riemann hypothesis). See [39]. Lat- Reminder. In a basic course on algebra we j n  ∞ q + 1  er, these definitions and questions where learn that for every prime power q = p there Z( 1 ,T ) = exp T j . PK X j considered for L-functions (not discussed exists, up to an isomorphism, exactly one j=1  here). field K with q elements; this field we denote In his description of the RH as Millenni- by K = . Note that /p, however Fq Fp ∼= Z This can be rewritten as um Problem Bombieri writes: “Not a single for n > 1 the field Fq is not isomorphic with n example of validity or failure of a Riemann hy- Z/p . 1 Z( 1 ,T ) = . pothesis for an L-function is known up to this PK (1 T )(1 qT ) date. The Riemann hypothesis for ζ(s) does From now on all base fields will be in char- − − not seem to be any easier than for Dirichlet acteristic p > 0. Surprisingly, the above examples give a ratio- L-functions (except possibly for non-trivial re- This theory, started by Emil Artin, gives rise nal function in the variable T . In an analogous al zeros), leading to the view that its solu- to the following definition, in analogy with way we can easily show: tion may require attacking much more gener- the for number fields. al problems, by means of entirely new ideas.” Let V be a non-singular, de- 1 [48] fined over K, i.e. an defined Z( n,T ) = PK (1 T )(1 qT ) (1 qnT ) d − − ··· − We seem to have made no progress for the as a closed set in some projective PK . classical RH in this way. However, we can de- For a field L containing K we write V (L) for the n rive some hope (or perhaps any hint?) from set of points on V with coordinates in L, the for all n Z>0 and all K = Fq. Note that P ∈ 0 n studying a special case of this more general set of points ‘rational over L’. We write can be written as disjoint union of A ,..., A ; situation: from this observation the formula follows. A Nm = Nm(V /Fq) := #(V (Km)), survey can be found in [25, Chapter 6]. Examples. For a given prime number p we study rings of finite type over the finite field the number of points on V with coordinates in Theorem (F.K. Schmidt, 1931). For a (non- singular, irreducible, projective) algebraic Fp (the prime field of characteristic p). Km := Kqm . This results in the (Hasse–Weil) s curve C over a finite field , its zeta func- 4. We take R = q, and obtain ζR = 1/(1 q ). zeta function Fq F − 5. Emil Artin proposed in 1924 in his PhD tion Z(C,T ) is a in T , having thesis [1] a definition of the zeta function Z(T ) = Z(V,T ) = ZV/K (T ) the precise form for an algebraic curve over a finite field  ∞ Nm m P and he proposed a RH for this kind of ze- = exp T . X m Z(C,T ) = , m=1  (1 T )(1 qT ) ta function. The definition and properties − − of such zeta functions were further studied by F.K. Schmidt [34] and by H. Hasse [18– Below we will see that this definition gener- where P [T ] is a polynomial of degree 2g, ∈ Z 19]. Rationality was proved by F.K. Schmidt alizes the notion of zeta functions ζR(s) as in where g = genus(C), and and Hasse proved the (analogue pRH of the previous section from rings to varieties. In the) RH for elliptic curves over a finite fact, Emil Artin introduced this zeta functions 2g P = P(T ) = (1 αiT ), αi . field. over a finite field, only for curves. Y − ∈ C i=1

From now on all varieties are defined over Example. Let V be just one point, rational

a finite field of characteristic p, and we study over K = q, hence Nm = 1 for every m > 0. The map α q/α is a permutation of F 7→ the pRH, an analog/special case of the ex- We see: α1, . . . , α2g . { } tended Riemann Hypothesis (below we give formulas). F.K. Schmidt proved in [34] the Riemann–  ∞ 1  1 Z(T ) = exp T m = . X m 1 T Roch theorem for curves in positive charac- m=1  Proof of the pRH by Hasse − teristic, and in the second part of that pa- In 1934 Hasse proves the pRH for elliptic per Schmidt uses this to prove the theorem d curves over finite fields. Example. Let us consider V = AK , the affine above, in particular the rationality of the zeta space of dimension d over K; this is de- function of a complete, nonsingular curve of Reminder. We write RH for the classical fined by: for any ring R containing K we have genus g over Fq, with numerator a polynomial d d Riemann Hypothesis (and generalizations in AK (R) = R . We see of degree 2g. characteristic zero). In order to avoid confu- d md sion we write pRH for the equivalent of the RH Nm = #((Fqm ) ) = q . Remark. We see that Z(C,T ) admits an ana- s about solving equations in positive character- lytic continuation. The substitution T = q− istic. We can show: gives

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s ZE (T ) = ZE (q− ) =: ζE (s). #(E( q)) 1 + q = α + β 2 q. I hope this convinces the reader that for any el- | F − | | | ≤ p liptic curve E over Fq after computing #(E(Fq)) We see that α q/α yields s 1 s, and See below. this theory gives access to all #(E(Fqm )). 7→ 7→ − 4 4 this is the (analogue of the) ‘functional equa- An amusing example: we see that α , β = (2√ 1)2 = 4; we obtain: tion’. Remark. As an elliptic curve is a double cov- − − er of the projective line (ramified in at least #(D(F16)) = 1 ( 4 4) + 16 = 25. Theorem (Hasse, 1934). For an elliptic curve one point) for an elliptic curve E over Fq we − − − E over K = q the algebraic integers α, β , immediately see the bound F ∈ C Here #(D( 16)) 1 16 = 8 = 2√16, a case defined by | F − − | #(E( q)) 2q + 1. where the Hasse bound is reached. F ≤ (1 αT )(1 βT ) However, note that any result of the type Z(E/K,T ) = − − (1 T )(1 qT ) pRH, the Riemann hypothesis in characteris- − − As 2√q < q + 1 for all q and 2√q < q for tic p, does not give any new result for the 4 < q we see the Hasse bound agrees with classical RH, for example: satisfy this bound and in many cases it is sharper. 1 (pRH) α = q = β . ζZ(s) = ζF (s) = | | p | | Elliptic curve Y p Y 1 p s p p − − We return to our example D 2 , the elliptic ⊂ PK Hasse proves in [18, 11] the Riemann Hy- curve given by the equation § (and it seems we have gained nothing in the pothesis for elliptic curves over a finite field, direction of the classical RH). see also [19, III, 4]. Y 2Z YZ2 = X3 X2Z § − − over the field K = F2. Remark. Instead of the definition given above Remark. The numerator P1(T ) of Z(E/K,T ) of Z (T ) one can give the equivalent defini- in case of an elliptic curve E is the eigenval- V tion: ue polynomial of the action of the Frobenius As we have seen we have #(D(F2)) = 5 (an π End(E), as Hasse remarked and used. easy computation, solving an equation mod- 1 ∈ ZV (T ) = , We will see that this insight will be general- ulo 2). By X 1 T deg(δ) δ ized. − #(D(F2)) = 5 = 1 (α + β) + 2, − where δ runs over all effective divisors of V Explanation. As already has been noted, we 2 2 β = , α + = 2, defined over . write RH for the classical Riemann Hypothesis, α α Fq and we write pRH for the special case of a gen- What is the structure behind such theo- eralization, studying an equivalent question we see rems, and how can these results be general- over finite fields. The property α = √q = β | | | | ized to curves of higher genus, and to vari- above implies that the zeros of ζE (s) have α, β = 1 p 1. 1 − ± − eties of higher dimensions? their real part equal to 2 here we see the re-

semblance with the classical RH. We conclude that for every m >0 we have ∈ Z Gauss In DA 358 [11] Gauss computes for certain We have seen that ZV (T ) encodes the se- m # (D(F2m )) = 1 ( 1 p 1) elliptic curve the number of rational points quence Nm m Z>0 . Conversely these − − ± − { | ∈ } m m over a prime field Fp; in the last entry in numbers can be retrieved via the logarithmic + ( 1 p 1)  + 2 . − ± − his Tagebuch [12], Gauss discusses these re- derivative (f ′/f ): sults as a conjecture. In [30] on page 73 (I find it astonishing that this theory after an G.J. Rieger writes: “Diese Tatsache wurde TZ (T ) ′ = N T m. easy computation gives this complete result.) ... übersehen und ist erst in neuere Zeit be- Z(T ) X m m>0 Let us see how this works out in simple merkt worden. Damit is auch die Richtighkeit examples. As the group D(F4) has D(F2) ∼= der Riemannschen Vermutung für denjenigen The Hasse bound /5 as a subgroup, and #(D( 4)) 2 4+1 = 9 Z F ≤ · Funktionenkörper nachgewiesen.” This claim It follows that for any elliptic curve E over Fq we conclude #(E(F4)) = 5 (this can also be puzzles me; it is not clear how computa- we have seen by an easy computation), and indeed: tions by Gauss imply the pRH in case you do not know rationality, functional equation 2 2 or something like that for the zeta function Nm(E/Fq) := #(E(Fqm )) α , β = 2 1, ∓ p− m m m 2 2 considered. = 1 (α + β ) + q . #(E( 4)) = 1 (α + β ) + 4 = 5. − F − Proof of the pRH by Weil We see that such a theorem gives the number Something you would not like to compute Andre´ Weil had the insight for the correct ap- of rational points over all finite extensions of without the preceding theory (α10, β10 = 15 proach to generalizations. He started to write ∓ K, once α and β are known. This proves that 8√ 1): foundations necessary for proofs. In the pe- − for any elliptic curve E over Fq we have the riod 1946–1948 he proved the pRH for alge- ‘Hasse bound’: #(D( 1024)) = 1 30 + 1024 = 995. braic curves of arbitrary genus and for abelian F −

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varieties. His proof starts with a simple, but rise to an abelian variety of dimension g, ory): abelian varieties over a finite field can fundamental observation. and many properties of C can be read off be constructed by just computing an algebra- n For a variety V over K = Fq with q = p the from properties of this abelian varieties. This ic integer having easy properties. map which sends coordinates xi of a point to abelian variety is called the Jacobian of C, or q xi is a morphism the Picard variety of C or the Albanese va- This theorem by Weil also proves the pRH riety of C; these notions coincide in case C for algebraic curves defined over a finite field. FrobV/K = πV/K : V V. → has at least one point rational over the base Reminder: a proof for the pRH for algebraic field. It is not deep (although it requires some curves defined over a finite field implies the Important (and obvious) remark. Fixed points work) that proving pRH for all curves amounts pRH for abelian varieties over a finite field. m of the map π : V V are exactly the Km = → to the same as proving pRH for all abelian va- Weil had two proofs. One proof relies m -rational points: Fq rieties. An abelian variety A is called simple on an analogue of a classical inequality if there is no non-zero proper sub-abelian va- (Castelnuovo-Severi) for correspondences on (fix πm) V (Fq) = V (Fqm ). riety 0 = B A. algebraic curves. Another proof (in this case   6 $ The terminology ‘abelian variety’ stems for abelian varieties) uses the result that the Let us compare this with the result (for an el- from the fact that Niels Henrik Abel studied Frobenius π = πA/K : A A for an abelian → t liptic curve) mentioned earlier: values of path-integrals of differentials on a variety over Fq and its ‘transpose’ π (the im- Riemann surface; their values naturally lie in age of π under the Rosati involution) satisfy m m m t t #(E( 2m )) = 1 (α + β ) + 2 . the related abelian variety, once you fix the π π = q. As π equals the complex conju- F − · end points of the path. gate of π, under any embedding ψ into C, we easily conclude ψ(π) = √q. What is an ‘interpretation’ of this formula, and | | how can we generalize this to arbitrary vari- Remark. An abelian variety A over K is In order to achieve such results Weil de- eties (defined over a finite field)? a group-object, the endomorphism algebra veloped many aspects of End(A) is a ring, and the Frobenius FrobA/K over an arbitrary field (in this case a field of ∈ Further explanation. For a projective algebra- End(A). In case A is simple, this ring has characteristic p). Note that Pierre Deligne was ic curve C over any field K one can define its no zero divisors, is of characteristic zero, and born in 1944. is finitely generated as a Z-module: its field genus, g(C) Z 0; for a curve over C there ∈ ≥ is a topological definition; in algebraic geom- of fractions is a number field (a field of fi- Corollary (The Hasse–Weil bound). For a etry this can be extended to curves over any nite degree over Q). Hence FrobA/K is (can non-singular, projective algebraic curve C of base field. A curve of genus zero is called a be considered) as an algebraic integer. Here genus g over K = Fq we have rational curve; a curve of genus one with a we see access to (pRH) in the case of abelian #(C(Fq)) 1 q 2g q. K- is called an elliptic curve. varieties, and hence in the case of algebraic | − − | ≤ p A projective, irreducible non-singular vari- curves. Here are a version and a corollary of results Indeed, #(C( q)) 1 q = αi and ety A such that this is a group variety is called F − − − Pi by Weil (1948): αi √q. an abelian variety. | | ≤ An abelian variety of dimension one is an elliptic curve. Any curve C of genus g gives Theorem. For a simple abelian variety A de- Remark. For elliptic curves, g = 1, we find fined over K = Fq its Frobenius homomor- this in [19, III, p. 206]. phism The Weil conjectures FrobA/K = π : A A → We will especially look into the RH for varieties over finite fields. is an algebraic integer, and under every em-

bedding ψ : (π) we have Reminder. For a curve C defined over K = q, Q → C F the pRH is just a statement on the asymptotic ψ(π) = q. p behavior of #(C( qm )) for m . | | F → ∞ As we have seen, for a variety V over n An amazing result: a ‘nature given’ endo- K = Fq, with q = p , the map which sends q morphism of a difficult object in positive char- coordinates xi of a point to xi is a morphism acteristic turns out to be an algebraic integer FrobV/K = πV/K : V V having easy properties. It is easy to charac- → terize and to produce such numbers (e.g. any 2 2 zero of T + bT + q with b Z and b < 4q; and V(Fqm ) is the set of fixed points of ∈ m many more examples can easily be given). FrobV/K .

Remark (details not explained nor discussed Andre´ Weil made in 1949 the following con- here). Conversely, an algebraic integer π with jecture: properties as in the theorem determines (an isogeny class of) an abelian variety A over Conjecture. Let V be a non-singular, projec- Andr´eWeil (1906–1998) Fq such that FrobA/K = π (Honda-Tate the- tive variety of dimension d defined over the

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finite field K = Fq having q elements. Then: the following disguise: 1. Rationality. The zeta function is a rational Show the graph of Frobenius is transversal s − function with coefficients in Z in T := q− : to the diagonal (and this is easy). Find some (co)homology theory (??!!) such − P1 P3 P2d 1 m k Z(V/K,T ) = × × · · · × − that the traces of (FrobV/K ) on H for all P0 P2 P2d k 0, 1, , 2 dim(V ) give the zeta- × × · · · × ∈ { ··· · } function of V /K. We should have Hk = 0 with for k < 0 and k > 2 dim(V ). · In the case of an algebraic curve this should d Pj [T ],P0 = 1 T,P2d = 1 q T. ∈ Z − − be: The trace of the Frobenius on the one- − 2. Analog of the classical Riemann Hypothe- dimensional H0 should be 1. m 1 sis. The polynomials P1, ,P2d 1 factor The trace of (FrobC/K ) on H should be ··· − − 2g over C as αm. Pi=1 i The trace of the Frobenius on the one- − 2 (pRH) Pk(T ) = (1 αk,j T ), dimensional H should be q. Y − j For an algebraic curve of genus g the degree k of P should be equal to 2g. For algebraic with αk,j = qq . 1 | | varieties of higher dimension the polynomials (1884–1972) P1,...,P2d 1 should be given in an analogous − Moreover there should be a functional equa- way. If so, we conclude (?!): tion, Poincare´ duality and an explanation of algebraic geometry with new insights and new

the degrees of the polynomials Pk in terms of conjectures. 0 1 #(C(Fqm )) = Tr f ∗ H  Tr f ∗ H  geometry of V : they should be (the analogues Here is that idea. We try to compute the | − | 2 + Tr f ∗ H of) the ‘Betti numbers’; if V is the reduction number of rational points on a variety V de-  |  mod p of a complex variety, these numbers fined over Fq over any finite field containing 2g = 1 αm + qm should be the in the complex- Fq. The zeta function encodes − X i topological sense. These together are called i=1 the ‘Weil conjectures’. Nm = #(V (Fqm )) for all m > 0. k k We have seen this conjecture to be true with H = H (X, ?) and f = FrobC/K . d for P , for algebraic curves and for abelian We have seen that Nm is the number of fixed I hope you see that earlier results by E. m varieties. The conjecture above is a daring points under the operator (FrobV/K ) . Artin, Schmidt, Hasse and Weil have a natu- generalization. How did Weil come to this In a completely different branch of math- ral, geometric interpretation and generaliza- insight? We will see that methods of algebraic ematics Lefschetz has indicated how to com- tion once you see this geometric approach, topology have their counterpart in arithmetic pute the number of fixed points of an opera- and once you find this elusive geometry. tor: theory. After this became clear, once you see it you Cohomology as predicted by Weil The Lefschetz fixed-point theorem (first stat- understand it, we “only” had to find this mys- Weil used in his proofs an interpretation ed in 1926). For a continuous map f : X X terious cohomology theory, and prove it has − → and generalizations of the notion of cor- from a compact triangulable space X to itself, the right properties. Then the Weil conjec- respondences as studied in the classical such that f has a finite set F = X(fix f ) of fixed tures would follow. Italian algebraic geometry. Also aspects points, and such that the graph of f intersects Completely independent from these ideas of the theory of abelian varieties were the diagonal in X X transversally the trace Dwork (1923–1998) proved in 1960 rationality × generalized to properties over arbitrary formula computes the number of fixed points: of the zeta function of a variety over a finite fields. field [9]. Also Monsky/Washnitzer and Lubkin k It might very well be that Weil originally #(F) = ( 1) Tr (f Hk(X, Q)) . contributed. X − ∗ | − k 0 had from the beginning a deeper motiva- ≥ tion behind his ideas. Alexandre Grothendieck In 1949 Weil stated his famous conjec- Let us compare this with the result for an In 1958 Grothendieck told us he was putting − tures, algebraic curve mentioned earlier: algebraic geometry on a new footing with the and in the ICM in 1954 he describes why main goal (for the time being) to prove the − these should be true, and what could be a 2g Weil conjectures, see [13]. I remember I was m m #(C( qm )) = 1 α + q . way to prove these. F − X i present at his presentation. I did not under- i=1 His idea how to proceed is of a great beauty, of stand his explanation, however there was one deep insight and of daring courage. Note that person in the audience who was very much Pierre Deligne graduated from high school in “Je gaat het pas zien als je het doorhebt”, as ‘au courant’; then I did not not know yet the Brussels in 1962. This insight by Weil started Johan Cruijff says. I would paraphrase: “Once influence Serre would have on mathematics a new chapter in arithmetic algebraic geom- you see it you understand it.” and also upon on me. Note that Pierre Deligne etry, a revolution, and a whole new setup of This is the Lefschetz fixed point formula in was 14 years old at that time.

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Serre made several attempts to construct of ℓ-adic integers respectively the field Qℓ of cause he knew now that the proof was roughly a ‘Weil cohomology’. His cohomology with ℓ-adic numbers. done.” See [32, p. 19]. values in the Witt vectors [35], 1958, did not For every variety V of dimension d over − bring this success. a field of characteristic p and for every A wonderful description of the proof to ap- His attempt to use ‘etale´ topology’ turned choice of ℓ there exists a cohomology pear later in [4] we find in [40]. We will not out to be fruitful. In Serre [36], 1958, we find theory H∗(V, Zℓ), the ‘etale´ cohomology’. — alas! — describe the beautiful proof of the idea. In order to have the analogue of fiber (As usual, obtained by derived functors in Deligne here. It seems much better either to spaces in algebraic geometry one should in- some topology, here in the ‘etale´ topology’ read the original paper by Deligne, or to go troduce a new notion of covering. Suppose of H0). through the description of Serre [40], or the G H is a homomorphism between algebra- For this cohomology properties like dual- description by Katz in [20]. → − ic groups (e.g. dividing out an elliptic curve ity and more other vital ingredients were Deligne knew aspects and difficulties of E = G by a finite subgroup, H = E/N). We proved. this problem inside out (“I had all these tools see this need not be locally trivial in the Zaris- From these Grothendieck and his co-workers at my disposal”). Instead of finishing of the ki topology (e.g. if G is irreducible and N is a derived: general program of Grothendieck on cycles finite group, but N = 0), but the covering can ( ) 6 Rationality of ZV T and hence analytic (still a complete mystery), Deligne followed a − s be trivialized (locally) by an etale´ map to a continuation of ζV (s) = ZV (q− ). suggestion by Serre: “I think it was Serre who Zariski open of H. Objects in this new notion A Lefschetz type of fixed point formula for told me about an estimate due to Rankin”; see − of ‘covering’ are maps with certain properties the Frobenius morphism. [29] (or came the suggestion from Langlands?, onto Zariski open subsets. We see that the Functional equation. see [20, p. 287]). This idea did put Deligne on − search for a proof can give new insights and See [14] for a first description of these results. the right track to finish the proof. However, constructions of radically new concepts. Proofs were described in various volumes of as Deligne says: “It would have been much This opened the way for Grothendieck to SGA. For example, see [10, 27–28, 39], [17, nicer if the program had been realized”, see define a new notion of topologies, and to Appendix C] and also see [38]. [32, p. 18] (i.e. the program by Grothendieck: construct the sought-for cohomology theory. However the property pRH, the character- the proof of the Weil conjectures via the stan- Grothendieck on page 104 of [13] seemed to istic p analogue of the Riemann hypothesis, dard conjectures). be the first who used the term ‘Weil cohomol- still escaped proofs: For a description of the prehistory see [33]

ogy’. (pRH) The eigenvalues of FrobV on (4 papers), [8, 26, 46], and references to E. − Grothendieck, together with , i i Artin, F.K. Schmidt and Hasse cited above. H (V, Zℓ) have absolute value qq . Here Giraud, Michel Raynaud, Illusie, Deligne and For a description of the Weil conjectures, see: the coefficient ring is Zℓ, the ring of ℓ-adic many others managed to find the earlier elu- integers, where ℓ is a prime number differ- [10, 26–27, 44] and [17, Appendix C]. Work sive ‘Weil cohomology’. They were able to ent from p. by Tate, see [41] can be seen as pre-history pin down basic properties. Here we record How to proceed? We see once the general of Grothendieck’s standard conjectures. For just some of these properties: fix the prime machinery of cohomology developed, most of a complete and beautiful survey of the pre- number p and choose another prime number the aspects of the Weil conjectures followed history, and of Deligne’s proof see [20]. For a ℓ = p. It turned out that the best choice for 6 by ‘pure thought’. However the analogue survey of Deligne’s proof, see [40]. a ring or field of constants was , the ring Zℓ (pRH) of the Riemann Hypothesis seemed out of reach of abstract theory. Pierre Deligne It is wonderful to see how various mathe- Maybe the essence of mathematics of Pierre maticians did continue in different ways. Deligne is described in his words: “Proofs in geometry make sense at that age because Generalization. Grothendieck came with surprising statements have not too difficult much more general conjectures; a survey of proofs.” These words are telling us about his these conjectures can be found in: [16, 21– interests in mathematics already at a very ear- 24]. These ‘standard conjectures’ on algebra- ly stage [32, p. 16]. I would like to see this as a ic cycles describe, amongst others, relations qualification for all of his mathematical work. between algebraic cycles and the Weil conjec- Throughout the years Deligne has amazed us tures. Once these would be proved the (pRH) by his clear insight and his beautiful and clear would simply follow. A fascinating idea. How- description of structures and proofs in math- ever most of this material seems completely ematics. inaccessible for the time being. We hope this note will give enough cred- it to all mathematicians on whose shoulders Inspiration from other fields. Deligne tried the theory was built (Euler, Riemann, Emil to prove the (pRH) directly (as many others Artin, Hasse, Weil, Serre, Grothendieck and had tried). Finally he was convinced he could many others) on the one hand, and on the prove the (pRH), Deligne wrote to Serre and other hand it will show the quantum leap, remembers: “I wrote him a letter ... I think he the deep insight of Deligne that was vital got it just before he had to go to the hospi- in this proof of the Weil conjectures (well,

Photo: Archive of Winfried Scharlau tal for an operation of a torn tendon. He told a ‘surprising statement’, however not an me later that he went in a euphoric state be- easy proof).

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problem or use the theory as a completely fa- miliar object.” See http://www-history.mcs. st-and.ac.uk/Biographies/Deligne.html I ful- ly agree with this remarkable description. In- deed, you have the feeling: Pierre Deligne is “able to discuss the problem or use the theory as a completely familiar object”. With this ability he came under the in- fluence of Serre (1926) and of Grothendieck (1928) (his official PhD-advisor) in his Paris years. Not only did he learn much from these two towering figures in algebraic ge- ometry, but the respect and admiration was certainly reciprocal. In their correspon-

dence Grothendieck complains to Serre that Pierre Deligne on a Belgian postage stamp his ‘anciens elèves’´ did not continue his work. Serre answers that this is not sur- the Weil conjectures implies the Ramanujan- prising: “You have the vision on a pro- Petersson conjecture holds true. E.g. see [25, gramme, and they do not have that (with Photo: CNRS 6.4.1]. k Pierre Deligne the exception of Deligne, of course)”, see [2, p. 244]. Pierre Deligne was born in 1944 in Belgium. In his Paris period Deligne has an impres- Already at a young age it was clear his in- sive production, not only in diversity of prob- Pierre Deligne sight and taste for deep mathematics could lems studied, but especially in depth of un- enable him to do extraordinary things. In the derstanding. Grothendieck tries to prove the Some of his many awards: interview with Pierre Deligne in the Newslet- Weil conjectures, and Deligne witnessed, and Abel Prize (2013) ter of the European Mathematical Society of collaborated at close range. In 1973 Deligne Wolf Prize (2008) September 2013 we obtain a clear picture of proves the last missing part, the Riemann hy- Balzan Prize (2004) this modest and friendly person, see [32]. pothesis (pRH), in the program of the Weil Crafoord Prize (1988) In the period Deligne was still in elementary conjectures. This is seen as his most pres- (1978) school the father of a friend gave him Bour- tigious achievement. For this he receives in baki mathematics to read. His (‘excellent ele- 1978 the Fields medal. In 1984 Deligne moved Timeline: mentary school’) teacher did put him in con- to the Institute for Advanced Study in Prince- 1944 Born tact with Jacques Tits. In high school Deligne ton. 1962 Finishes High School enjoys problems in geometry (see the citation Let me mention here one of his theorems: 1966 Finishes University (Free University above). The Ramanujan-Petersson conjecture. This Brussels) Deligne at age 16 joins a course by Jacques is a statement about Fourier coefficients of 1968 PhD University of Paris-Sud (France) Tits, who comments: “A remarkable feature an interesting (no explanation 1972 Doctorat d’Etat´ des Sciences Mathe´ of Pierre Deligne’s thinking is that, when con- given here). Ramanujan conjectured their matiques fronted with a new problem or a new theo- absolute values should satisfy inequalities 1968–1970, 1970–1984 IHES Bures sur ry, he understands and, so to speak, makes as printed on a Belgium stamp. In 1971 Yvette (France) his own its basic principles at a tremendous Deligne proved this would follow from pRH. 1984 IAS Princeton (USA) ≥ speed, and is immediately able to discuss the Hence the proof (1974) of the aspect pRH of

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