Lecture 1: Weil conjectures and motivation

Transcription

1 Lecture 1: Weil cojectures ad motivatio September 15, The Zeta fuctio of a curve We begi by motivatig ad itroducig the Weil cojectures, which was bothy historically fudametal for the developmet of Etale cohomology, ad also costitutes oe of its greatest successes. It has log bee kow that there is a strog aalogy betwee rigs of itegers i umber fields, ad smooth projective curves over fiite fields. As such, let us begi with the usual Riema Zeta fuctios. The Riema zeta fuctio is most commoly defied as follows: for R(s > 1, ζ(s = =1 1 s. (1 Usig the uique factorizatio theorem, we ca also rewrite the above sum as a product: for R(s > 1, ζ(s = p (1 p s 1 ( The Riema Zeta fuctio ejoys the followig properties: While oly defied iitially for R(s > 1, ζ(s ca be mermorphicaly cotiued to the etire complex plae, with oly a simple pole at s = 1. There is a fuctioal equatio satisfied by ζ(s, give by ζ(sγ( s s π = ζ(1 sγ( 1 s πs 1. (Riema Hypothesis: Oly cojectural! The zeroes of ζ(s all lie o the lie R(s = 1, with the exceptio of the trivial zeroes that occur at the egative eve itegers. 1

2 Remark. The above suggests that the fuctio ξ(s = ζ(sγ( s s π is more atural to work with the the Zeta fuctio, as it satisfies a icer fuctioal equatio ξ(s = ξ(1 s ad elimiates the trivial zeroes. The reaso for this is that it is atural to cosider the Archimedea prime at i the product formula (, ad it turs out that Γ( s s π is the atural factor at that prime. We will ot go ito the justificatio of this heuristic, which ca be foud withi Arakelov theory or the theory of automorphic forms. To try ad make a aalogy with fiite fields, we thik geometrically. Thus we form the scheme spec Z. The closed poits of spec Z are precisely give by the prime ideals of Z, which are i bijectio with the primes. Thus, the closed poits are simply spec F p spec Z, ad the prime umbers p are simply the sizes of the residue fields spec F p. Now, we are ready to formulate a geometric aalogue. Let q be a prime power, ad X a smooth, projective curve over F q. What do the closed poits of X look like? Well, each closed poit x X has residue field some fiite field of the form F q. Let us write deg(x = ad N(x for q, the size of the residue field k(x at X. Thus, we make the followig defiitio: ζ(x, s := x X(1 N(x s 1. (3 We see that this defiitio is exactly aalogous to (. What about the represetatio as a sum as i (1? The aalogous otio of a iteger here is that of a positive divisor, which o a curve is just a fiite formal sum of poits with o-egative coefficiets. For D = i a ix i, we defie N(D = i N(x i a i. Expadig the product as with the Zeta fuctio, we get ζ(x, s = N(D s. D Sice we are ow i the world of geometry, we ca also rewrite the Zeta fuctio i a third way, by coutig poits i field extesio; that is, usig the quatities X(F q. Specifically, if deg(x = d, the x cotributes d poits to X(F q if d, ad o poits otherwise. Geometrically, oe ca thik of it as follows: a poit y X(F q is a map y : spec F q X. The image of y is some poit x X, ad thus we ca factor the map as spec F q spec k(xspec X. Now, a map from spec F q to spec k(x is by defiitio a embeddig of fields k(x F q, ad sice all fiite field extesios are Galois, there are either N(x such extesios if d or 0 otherwise.

3 Usig the power series expasio for log, we ca ow write log ζ(x, s = x X d=1 = = N(x ds =1 N(x =1 d N(xq s X(F q q s Expoetiatig, we have Z(X, s = exp =1 X(F q q s. (4 So the Zeta fuctio also records the umber of poits of a variety i extesio field, ad these are extraordiarily iterestig. Let us do a example. Cosider the case of X = P 1 /F q. We see that X(F q = q + 1, sice we have q elemets (1 : t with t F q together with the poit at ifiity (0 : 1. Thus, usig (4 we calculate Z(X, s = exp = exp =1 =1 q + 1 q s q (1 s exp = (1 q 1 s 1 (1 q s 1. =1 q s For X a curve of higher geus, it is o loger so easy to cout poits. Oe might woder how to eve proceed with computig the Zeta fuctio. It turs out the Riema-Roch formua ca help. Recall equatio (. Now a divisor D gives us a lie budle l(d.moreover, give a lie budle l, it has H 0 (X, l 1 may sectios, ad up to the actio of F q they each give a differet divisor givig rise to l. Thus, writig Pic(F q for the set of lie budles of degree, we ca rewrite 3

4 Z(X, s = 0 q s l Pic ( F q H 0 (X, l 1. q 1 Moreover, by the Riema-Roch theorem, if g X 1 the H 0 (X, l = q g+1. Thus, we ca write the above as Z(X, s = P (q s (1 q s (1 q 1 s where P (T is a polyomial of degree g X. Moreover, it turs out that Z(X, s satisfies a fuctioal equatio Z(X, s = ±Z(X, 1 sq (1 gx(1/ s, ad by a theorem of Weil, all the roots of P (T have absolute value q 1/, which traslates to the zeroes of Z(X, s all beig o the lie R(s = 1 ; that is, the Riema hypothesis holds! At this poit, it is atural to ask what happes if we go to higher dimesios. So suppose X is a smooth, projective variety over F q. The we ca defie the Zeta fuctio of X exactly as i (3. Moreover, by the same aalysis, this will be idetical to the represetatio i (4 1.The oe hiccup is that divisors are o loger collectios of poits, ad so the represetatio ( is o loger applicable. As a example, oe ca compute the zeta fuctio i the case X = P to be 1 Z(X, s = (1 q s (1 q 1 s... (1 q s. Statemet of the Weil Cojectures At this poit we are ready to state the Weil cojectures. These were Made after Weil after he computed a plethora of examples - a feat i itself, as computig poits over fiite fields is ote easy. Theorem.1. (Weil Cojectures Suppose X is a smooth projective variety of dimesio over F q. The the Zeta fuctio of X satisfies the followig properties: 1. (Ratioality The Zeta fuctio Z(X, s is a ratioal fuctio of q s. 1 To avoid cofusio, let me clarify that this equality has othig to do with either the smoothess or the projectivity assumptio. Of course, we ca just replace the word divisor with 0-cycle ad it will hold. However, the Riema-Roch theorem is o loger applicable, ad so this represetatio is less useful. 4

5 . (Fuctioal equatio there is a iteger E such that Z(X, s = ±q E(/ s Z(X, s. 3. (Riema Hypothesis The Zeta fuctio ca be writte as a alteratig product Z(X, s = P 1(q s P 3 (q s... P 1 (q s P 0 (q s P (q s... P (q s where each P i (T is a itegral polyomial all of whose roots have absolute value q m/. Moreover, P 0 (T = 1 T ad P (T = 1 q T. 4. (Betti Numbers Suppose X is a good reductio of a characteristic zero variety. That is, there is a smooth projective morphism X Y such that the base chage w.r.t oe of the spec F q -valued poits of Y is X, ad the base chage to oe of the spec C-valued poits of Y is a smooth projective complex variety X 0. The the degree of the i th polyomial P i is the i th betti umber of the space of the topological space Y (C. Note i particular the Riema Hypothesis - called such because it places the zeroes ad poles of Z(X, s o ice vertical lies i the complex plae. The weil cojectures, as we sketch ext sectio, led to the developmet of Etale cohomology, as (4 above suggests that a certai cohomology theory is lurkig i the backgroud, ad Grothedieck realized that a suitable cohomology theory would be very useful i provig the Weil cojectures. We should metio that the ratioality of the Zeta fuctio was first prove by Dwork before the developmet of Etale cohomology, though his proof did ot give early as much iformatio. 3 Cohomology of maifolds ad Grothedieck s Dream Let s recall how ordiary topological Cech cohomology works, ad the we ll see why a appropriate aalogue would be useful i provig the Weil cojectures. So suppose M is a -dimesioal compact real maifold, ad T is a triagulatio of M ito simplices. Let T i be the i-dimesioal simplices i T.Let C i deote the set of maps from T i to Q. Fially, let d m be the map 5

6 C m C m+1 defied as follows: d m (φ(v 0, v 1,..., v m = m ( 1 i φ(v 0,..., v i 1, v i+1,..., v m. The it is easy to verify that d i+1 d i = 0, ad so we get a complex d C 1 d 0 C1 C... d 1 C. The we defie the Cech Cohomology groups to be H i (M, Q := ker d i /imd i+1. It is true (though ot obvious that give ay two triagulatios of M, their cohomology groups ca be aturally idetified. Moreover, we have the followig woderful properties: The groups H i (M, Q are fiite dimesioal. Moreover, if M is a complex algebraic algebraic variety, the H j (M, Q = 0 for j > dim C M. (Fuctoriality For ay cotiuous map φ : M N, we have iduced maps φ i : H i (M, Q H i (N, Q compatible with compositios. (Poicare Duality The groups H i (M, Q ad H i (M, Q are caoically dual. Moreover, H (M, Q is oe dimesioal, ad there is a atural perfect pairig H i (M, Q H i (M, Q H (M, Q. (Lefschetz trace formua Suppose φ : M M is a cotiuous map with oly simple, 3 isolated fixed poits. The #{fixed poits of φ} = ( 1 i tr(φ i. Now, suppose for a secod that we had a way to defie a cohomology theory for proper,smooth varieties X over fiite fields satisfyig some versio of the above properties. The reaso this is useful, is that if X is a variety over F q, the we have a atural map X X kow as the Absolute Frobeius morphism. If X = spec A the this is iduced by the map of rigs A A give by 4 a a q, ad otherwise its defied by gluig 5. The it is ot hard to see that the fixed poits of F m o X(F q are exactly X(F q m. So we 3 This is a bit techical to defie. But if M is a smooth maifold the its eough to say that the graph of φ i M M is trasverse to the diagoal 4 Verify this is a map of rigs! 5 Check that this glues! 6

7 could hope that some versio of the Lefschetz Trace formula would imply that #X(F q m = dim X ( q i trf H i (X,Q. Combiig this with the formal idetity of matrices log det(1 T M = trm i i=1 i we would deduce that Z(X, s = dim X det(1 q s F H i (X,Q. This would imply the ratioality of the Zeta fuctio immediately. Moreover, oe ca see that a appropriate versio of Poicare Duality would yield the fuctioal equatio, ad the compatibility with reductio from characteristic 0 would follow from some sort of compatibility with regular cohomology. This also strogly suggests that the sought for polyomials i (3 of.1 are P i (T = det(1 T F H i (X,Q ad reformulates the Riema Hypothesis as sayig that the eigevalues of F o the i th cohomology group are of size q i/ - this is the oly part of the Weil cojectures that would ot follow formally from the Weil cojectures, but it still provides some isight ito whats goig o 6.This sketch is what we will justify usig Etale Cohomology. It turs out that we caot have our coefficiet group be Q we eed to use a profiite group such as Q l but the basic ideas remai the same. As a fial commet, we poit out that i tryig to defie a cohomology theory to satisfy all of the above, the Zariski topology is grossly iadequate. For istace, the Zariski topology o ay two curves is idetical(prove this! 6 Ad is essetial to Delige s evetual resolutio of the Riema Hypothesis 7