Frans Oort The Weil conjectures NAW 5/15 nr. 3 september 2014 211 Frans Oort Mathematisch Instituut Universiteit Utrecht [email protected] Geschiedenis The Weil conjectures 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 Pierre Deligne. 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 conjecture 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 mathematics’ 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. Nick Katz in [20] is negative in the strict sense: there is no im- plication pRH RH (and also no argument the ⇒ The Riemann Hypothesis (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 Emil Artin, 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 1 1 212 NAW 5/15 nr. 3 september 2014 The Weil conjectures Frans Oort number of rational points on several elliptic Euler series and the Riemann zeta function 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 elliptic curve E over a field K is a non- ler, already in 1740, for positive integral values 1 ζZ(s) = 2 Y s singular algebraic curve 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- 2 2 Frans Oort The Weil conjectures NAW 5/15 nr.