1 Counting points with Zeta functions

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1 The goal of this lecture is to preset a motivatio ad overview of the étale ad pro-étale topologies ad cohomologies. 1 Coutig poits with Zeta fuctios We begi with the followig questio: Questio 1. Let X is a smooth projective variety over F q, how may elemets does the set X(F q ) = hom Spec(Fq)(Spec(F q ), X)) of F q -poits of X have for each? Or equivaletly: Questio 2. If f 1,..., f c F q [t 0,..., t d ] are the homogeeous polyomials defiig X, how may solutios do f 1,..., f c have i F q for each? I order to work with all the sets X(F q ) at oce, we itroduce the zeta fuctio ( ) Z(X, t) = exp X(F q ) t ( ) = ( 1 t deg(x)) 1. =1 x X Exercise 1. Applyig log ad usig the idetity log(1 T ) 1 = =1 T /, prove the equality ( ). Remark 3. Note Z(X, t) is defied for ay F q -variety, possibly ot projective, ot smooth. Remark 4. For ay sequece of closed subsets Y 0 Y 1 Y = X, it follows from the sum descriptio that we have Z(X, t) = i Z(Y i \ Y i 1, t). Remark 5. There is a reaso that the product form of Z(X, p s ) looks similar to the Riema zeta fuctio ζ(x, s) = p prime (1 p s ) 1 but we will ot say much about this. Now our questio has become: Questio 6. Calculate Z(X, t). Example 7. First cosider Z(A d, t). We have A d (F q ) = q d so Z(A d, t) = exp (q d t) / = exp( log(1 q d 1 t)) = (1 q d t). =1 Example 8. Cosider X = P d. Choosig coordiates gives a sequece P 0 P 1 P d. Sice A i = P i \ P i 1, we see that Z(P d, t) = 1 (1 t)(1 qt)... (1 q d t) 1

2 Example 9. Let X be a elliptic curve. Usig the actio φ l : T l E T l E of the Frobeius φ o the Tate module T l E = lim ker(e l E) oe ca calculate E(F q ) = deg(1 φ ) = det(1 φ l ) = 1 α β + q where α, β C are complex cojugates with absolute value q. The usig the log argumet as i the case of A d, we fid that Z(E, t) = (1 αt)(1 βt) (1 t)(1 qt). For more details see Silverma, The Arithmetic of Elliptic Curves, Chapter 5. This method geeralises to higher dimesio abelia varieties. Example 10. If X is a curve, the the Zeta fuctio ca be rewritte i terms of divisors, ad from there, i terms of liear systems of divisors of lie budles. The usig the Riema-Roch theorem for curves, oe ca calculate Z(X, t) = f(t) (1 t)(1 qt) where f(t) Z[t] has degree 2g. For more details see, for example, Raski, The Weil cojectures for curves. Example 11. Usig characters χ : F q C, oe ca calculate explicitly the case X is a smooth hypersurface defied by a equatio of the form a 0 x a 1 x a rx r r. Z(X, t) = 1 (1 t)(1 qt)... (1 q r 1 T ) (1 C(α)t µ(α) ) ( 1) where α (F q) r+1, µ(α) N, C(α) C, C(α) = q (r 1)µ(α) 2, ad we do ot say what the product is over. For details see Weil, Numbers of solutios of equatios i fiite fields. After calculatig may examples Weil made the followig cojectures: Theorem 12 (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, t) is a ratioal fuctio of t. 2. (Fuctioal equatio) There is a iteger e such that Z(X, q t 1 ) = ±q e/2 t e Z(X, t). 3. (Riema Hypothesis) The Zeta fuctio ca be writte as a alteratig product Z(X, t) = P 1(t)P 3 (t)... P 2 1 (t) P 0 (t)p 2 (t)... P 2 (t) where each P i (t) is a itegral polyomial all of whose roots have absolute value q i/2. Moreover, P 0 (t) = 1 t ad P 2 (t) = 1 q t. α r µ(α) 2

3 4. (Betti umbers) Suppose there is a umber field K/Q, ad homogeeous polyomials f 1,..., f c O K [t 0,..., t d ] where O K is the rig of itegers of K, such that X is defied by the f i mod p, ad the complex projective variety X(C) defied by the f i is smooth. The deg P i (t) = dim Q H i (X(C), Q) where X C P d C is give the topology iduced from Pd C cosidered as a complex aalytic space. Remark 13. The Riema Hypothesis is so called because it places the zeroes ad poles of Z(X, q s ) o vertical lies i the complex plae. Exercise 2. Show that if s is a zero or pole of Z(X, q s ) the Rs = j/2 for some j Z. 2 Coutig poits with cohomology Now we show why oe might expect cohomology to be useful. Suppose M is a -dimesioal compact real maifold. The sheaf cohomology H i (M, Q) of the costat sheaf Q has the followig properties. 1. (Fiiteess) dim Q H i (M, Q) < for all i. Moreover, if M = X(C) comes from a complex algebraic variety X, the H i (M, Q) = 0 for i > 2 dim C X(C). 2. (Fuctoriality) For ay cotiuous map φ : M N, there are iduced maps φ i : H i (M, Q) H i (N, Q) compatible with compositio. 3. (Poicaré Duality) There is a caoical isomorphism H (M, Q) = Q, ad a atural perfect pairig H i (M, Q) H i (M, Q) H (M, Q). 4. (Lefschetz Trace Formula) Suppose φ : M M is a cotiuous map with oly simple isolated fixed poits (e.g., the graph is trasverse to the diagoal). The #{ fixed poits } = ( 1) i tr(φ i ). Now suppose we had a cohomology theory defied for algebraic varieties over fiite fields, satisfyig versios of the above properties. Sice X(F q m) = fixed poits of F rob m : X(F q ) X(F q ) we could hope that a versio of (Lefschetz Trace Formula) would give X(F q m) = 2 dim X 3 ( 1) i trφ m i

4 with φ = F rob. Isertig this to the sum descriptio of Z(X, t) we get ( 2 dim X ) Z(X, t) = exp ( 1) i trφ t i Combiig this with = valid for ay matrix A, we get =1 2 dim X i=1 ( exp trφ t i =1 det(1 A) 1 = exp Z(X, t) = 2 dim X tra / =1 ) ( 1) i det(1 φ i t) ( 1)i+1 ad we would get (Ratioality). Moreover, a appropriate versio of (Poicaré Duality) would give (Fuctioal equatio), ad if our ew cohomology theory is compatible with usual cohomology i a appropriate way, the we would get (Betti umbers). Fially, this descriptio suggests that the polyomials i (Riema Hypothesis) are P i (t) = det(1 φ i t), ad if so, the the secod part is reformulated as: the eigevalues of φ i have absolute value q i/2. 3 Fudametal group Notice that i the Zariski topology, every o-empty ope subsets of a smooth coected variety is dese. Cosequetly, H (X, Q) = 0 for all > 0. So we eed a more sophisticated cohomology. Leavig cohomology aloe for a momet, lets cosider the fudametal group. Note that for smooth maifolds M, there is a caoical morphism π k (M) H k (M) ad if M is path coected the the Hurewicz Theorem says π 1 (M) H 1 (M), [π 1 (M), π 1 (M)] which is dual to H 1 (M) whe M is compact ad orieted. So if we ca fid a good algebraic versio of the fudametal group, this might give some idicatio how to build a cohomology theory. Recall: Defiitio 14. The fudametal group of a smooth maifold M is π 1 (M, m) = hom cot. ([0, 1], (M, m))/ hom cot. ([0, 1] 2, (M, m)). 4

5 That is, loops from m to m, modulo homotopy. picture of a loop ad a cotractio The first problem with this defiitio i the world of varieties is that [0, 1] is ot algebraic. We could observe that extedig [0, 1] A 1 (C) gives the same groups, ad try ad use A 1. But this is still o good. Topologically, for a complete elliptic curve E, we have E(C) = S 1 S 1, but every algebraic map A 1 E is costat. Lets use a differet descriptio of the fudametal group. Theorem 15. Let M be a smooth (coected) maifold, ad M M a smooth morphism of relative dimesio zero such that M is cotractible. The π 1 (M) = Aut( M/M). picture of S1, uiversal cover, ad a loop goig to a automorphism Smooth morphisms ca be defied algebraically, but cotractibility caot (yet). However, if we do t mid passig to the completio, we are ok. Theorem 16. Let M be a smooth (coected) maifold, ad cosider the category of all fiite, smooth, relative dimesio zero morphisms N M. π 1 (M) = lim Aut(N/M). N M picture of S1, uiversal cover, ad a loop goig to a automorphism This leads to a useful otio of fudametal group. Defiitio 17. Let X be a smooth variety ad cosider the category of fiite, smooth, relative dimesio zero morphisms Y X. Defie 1 (X) = lim Aut(Y/X). π et Y X A cosequece of the Riema Existece Theorem is the followig. Theorem 18. Let X be a smooth C-variety. The π et 1 (X) = π 1 (X(C)) Moreover, this étale fudametal group cotais arithmetic iformatio. Propositio 19. Let k be a field with algebraic closure k. The π et 1 (k) = Gal(k/k) 5

6 4 Cohomology via local homeomorphisms Remark 20 (This remark may ot appear i the lecture). What does the Hurewicz Theorem look like ow? A fiite smooth morphism N M is called Galois if Aut(N/M) = deg N. I this case there is a caoical 1 isomorphism N M N = Aut N. Let Λ be ay abelia group. Whe N is coected, set morphisms N M N Λ are i bijectio with global sectios of the costat sheaf Γ(N M N, Λ). Oe ca check that a fuctio N M N Λ correspods to a group homomorphism φ : Aut(N) Λ if ad oly if the correspodig sectio σ φ Γ(N M N, Λ) is a cocycle. Moreover, the morphism φ is trivial if ad oly if the sectio σ φ comes from Γ(N, Λ). So we see that hom(aut(n), Z/) = Ȟ 1 (N/M, Λ), where Ȟ (N/M, Λ) is the cohomology of the complex associated to N M N M N N M N N. Note how similar this looks to Čech cohomology, which for a coverig {U i M} i I is the complex associated to U M U M U U M U U where U = U i. All of this suggests that we should be workig with smooth relative dimesio zero morphisms. Defiitio 21. A smooth relative dimesio zero morphism f : Y X is called étale. To give more support for this idea, otice that the iverse fuctio theorem ow holds: Propositio 22. Let f : Y X be a smooth morphism of varieties. The for every y Y, there exists a étale morphism j : U X, a poit u U with j(u) = f(y), ad a factorisatio Y U X Exercise 3. Prove the above propositio whe Y X is of relative dimesio zero. Moreover, just as locally every maifold looks like R, étale locally every smooth variety looks like A : Propositio 23. If X is a smooth variety of dimesio, the for every x X, there exists a ope subset U x, ad a étale morphism U A. Exercise 4 (Advaced). Prove the above propositio whe X is of dimesio oe. Hit: use the fact that each O X,x is a discrete valuatio rig. 1 The diagoal N N M N is a caoical choice of coected compoet, ad the Aut actig o the right of N M N permutes the compoets. 6

7 The above suggest that we should cosider étale morphisms as uwrapped ope subsets. Now otice that the defiitio of a sheaf oly the structure of the poset of ope sets, ad othig to do with the fact that they are subsets: Defiitio 24. A fuctor F : Ope(M) op Ab is a sheaf, if for every ope subset V M ad every ope cover {U i V } i I the sequece 0 F (V ) F (U i ) F (U i V U j ) i I i,j I is exact. Fially, we arrive at the defiitio of a étale sheaf. Defiitio 25. Let X be a scheme, ad Et(X) the category of étale morphisms V X. A fuctor F : Et(X) op Ab is a étale sheaf, if for every V Et(X), ad every joitly surjective family {U i V } i I i Et(X), the sequece 0 F (V ) F (U i ) F (U i V U j ) i I is exact. i,j I 7