Unramified CFT for proper smooth varieties APRAMEYO PAL November 13, 2014 Abstract We will study unramified class field theory for higher dimensional proper smooth varieties. 0 Classical CFT Let X be a smooth projective curve and K = F q (X) be the global function field. Let A K be the idele group of K. From Xevi s talk, we have the following classical reciprocity law, A K θ K Gal(K ab /K) res Gal(K ur,ab /K) Here A K = K v = (O v π Z ) = (O v Z) and the local Artin map sends O v to the inertia group at v. So res θ K sends O v to 1 as the target group is unramified. We get the Global Artin map, By reciprocity law, this induces θ K : x X Z Gal(K ur,ab /K) θ K : coker(k x X Z) Gal(K ur,ab /K) by definition, Chow group of zero cycles := CH 0 (X) coker(k x X Z) and π 1 (X) ab Gal(K ur,ab /K). So we get the reciprocity map CH 0 (X) θ K π1 ab (X)... ( ) This induces an isomorphism between the finite groups ker(ch 0 (X) deg Z) := CH 0 (X) π ab 1 (X) := ker(π ab 1 (X) π ab 1 (F q )) The AIM of today s talk is to generalize (*) to higher dimensional varieties. 1
1 Higher dimensional CFT Theorem 1.1. Let X be a connected, projective and smooth variety over k = F q. Then the reciprocity map is injective and we have a commutative diagram of exact sequence 0 CH 0(X) CH 0(X) deg Z 0 φ X 0 π ab 1 (X) π ab 1 (X) Ẑ 0 where CH 0 (X) := subgroup of CH 0 (X) consisting all cycle classes of degree zero and π1 ab (X) := ker(π1 ab (X) Gal( k/k) Ẑ). We will first define CH 0 (X) and the reciprocity map for higher dimensional varieties. Steps of the Proof: 1. π ab 1 (X) is finite (Katz-Lang). 2. image(φ x ) is dense in π1 ab (X) and φ X is surjective (Lang). 3. Define λ : CH 0 (X) π1 ab (X), when X is a surface and prove λ is injective. A priori, λ may not be same as φ X but they actually are same ( Colliot-Thélene-Sansuc-Soule). 4. Show λ and φ X are isomorphisms by counting argument (Colliot-Thélene- Raskind). 5. φ X is an isomorphism for any dimension by reducing X to a surface by using a hyperplane argument (Kato-Saito). (1) follows from Niels talk. In this talk, we will prove (2). 2 Chow group and Reciprocity map Let X be a connected smooth projective variety over a finite field k = F q (where q = p e ) and π1 ab (X) denotes the abelianized fundamental group of X (π1 ab (X) classifies unramified coverings of X). We define Z 0 (X) := Group of 0-cycles on X. Let x X be a closed point, then the residue field k(x) is a finite field. The absolute Galois group G x = Gal(k(x)/k(x)) π1 ab (spec(k(x)) Ẑ has a canonical generator, the frobenius ϕ x defined by ϕ x (a) = a N(x), for every a k(x) where N(x) = k(x). (G x π1 ab (X) can be considered as decomposition group of x, this is well defined as two decomposition groups are conjugate). We have a canonical map i x : spec(k(x)) X and push-forward (i x ) : π1 ab (spec(k(x)) π1 ab (X). So we can define (extending linearly) x X Z.x =: Z 0 (X) π1 ab (X) 2
by sending x n x.x x n x.(i x ) (ϕ x ). is called the reciprocity map. Our aim is to understand and π ab 1 (X). The first important theorem in this direction is the following (Due to Lang) Theorem 2.1. Let X be a normal variety over k, then has dense image in π ab 1 (X). Remark 2.2. As is dense, under mild technical condition, we hope to exhaust information regarding π1 ab (X) by algebraic loops. So we have to find appropiate homotopy relation among these loops which is to find elements in ker( ) with geometric interpretation. Hope is to obtain a map almost isomorphism after finding many relations and diving them out. Remark 2.3. Since Z 0 (X) is discrete and π1 ab (X) carries a natural compact topology, it will not be possible to find an isomorphism between a quotient of Z 0 (X) and π1 ab (X), unless π1 ab (X) is finite. In general case, should induce an isomorphism on profinite completion. Required equivalence relation = rational equivalence to 0, which will give us the definition of Chow group. Definition 2.4. A zero cycle D Z 0 (X) is rationally equivalent to 0 if there exists an integrally closed subscheme C X of dimension 1 and a rational function f on the normalization ψ : C C such that D = p (Div(f)) where p : C X is the natural morphism and p is induced by sending P C zero cycle [k(p ) : k(ψ(p )]ψ(p ). Definition 2.5. The quotient of Z 0 (X) modulo the subgroup of zero cycles rationally equivalent to 0, is called Chow group of zero cycles on X and denoted by CH 0 (X). Definition 2.6. (Alternative Definition) Let X be a noetherian scheme. For every integer j we put X j := (x X dim({x}) = j). We define the Chow groups by CH j (X) := coker( y Xj+1 k(y) y Xj Z) The maps are given by { 0 for x / {y} f y ord O{y},x f y for x {y}. They are well defined homomorphisms as dimo {y},x = 1 in the last case. Theorem 2.7. Let X be a proper smooth variety over k, then through rational equivalence to induce This is also called reciprocity map. : CH 0 (X) π ab 1 (X) factors 3
Proof. Using the definition 2.4 and the construction of, we have the following commutative diagram, Z 0( C) Z 0(C) Z 0(X) φ C φ C π ab 1 ( C) π ab 1 (C) π ab 1 (X) Thus we can reduce the problem to the 1-dimensional case, where it is true. So we are done. So we get the following commutative diagram 0 CH 0(X) CH 0(X) deg Z φ X 0 π ab 1 (X) π ab 1 (X) Ẑ 0 Definition 2.8. Let X be of finite type over k. Define ζ X by the Euler product ζ X (s) = where NP is the cardinality of k(p ). P X 1 (1 NP ) s Theorem 2.9. Let X be a separtaed scheme of finite type over k. Put d := dim(x) and r X :=cardinality of {T X 0 dim(t ) = dim(x)}, where X 0 is the set of irreducible components. Then ζ X is a rational function of p s with rational coefficients and it is holomorphic on {s C Re(s) > d 1} d and has a pole of order r X at s = d. Theorem 2.10. (Theorem 2.1) Let X be a normal variety over k, then has dense image in π1 ab (X). In other words, let χ : π1 ab (X) Q/Z be a continuous character, then if χ(ϕ x ) = 0 for all x X, then χ = 0. Proof. Assume H =< ϕ x > where x X and H be its closure. Let U := ker(χ). This is an open subgroup of π1 ab (X) containing H. This corresponds to a connected finite étale cover Y X (Galois with abelian galois group). Let P X, then the fibre Y P above P must be the spectrum of a finite direct product of copies of k(p ) as by construction of Y, ϕ P acts trivially on each geometric point of Y P. This implies Y has exactly d closed points lying above each closed point of X where d =degree of the cover Y X. Then by definition ζ Y = ζx d. Since ζ X and ζ Y have pole at order 1 at s = dim(x) = dim(y ) correspondingly, this implies d = 1. But then Y = X and hence U = π1 ab (X). By Galois theory, then χ = 0. Corollary 2.11. Let X be a proper geometrically connected integral variety over k then the degree map CH 0 (X) Z is surjective. 4
Proof. Consider the following commutative diagram Z 0(X) π ab 1 (X) Z deg f G k Ẑ As X is geometrically connected, f is surjective. Let nz = im(deg). But as has dense image so is true for f. But nz is dense in Ẑ if and only if n = ±1. So deg is surjective. Corollary 2.12. φ X is surjective. Proof. φ X is dense and the target group is finite so it is surjective. 3 Future directions Till now we consider unramified class field theory as π ab 1 (X) classifies unramified coverings of X. But if we allow ramification, then we can also do class field theory in similar spirit. These will be the topics of the following lectures. Let π1(x) t denotes the tame fundamental group classifying tame finite étale coverings of X. Then denote π t,ab 1 (X) the tame part of the abelianized fundamental group ( quotient in a natural way). Then we have following version of Theorem 1.1. Theorem 3.1. (Schmidt-Spiess) Let X be a smooth connected variety over k. Then the following reciprocity map : H0 S (X, Z) π t,ab 1 (X) gives the commutative diagram 0 H S 0 (X, Z) H S deg 0 (X, Z) Z 0 φ X 0 π t,ab 1 (X) π t,ab 1 (X) Ẑ 0 Remark 3.2. is defined similarly as above as a map : Z 0 (X) π1 ab (X) π t,ab 1 (X). This factors through Homotopy equivalence to give the suslin cohomology H0 S (X, Z). Remark 3.3. If X is proper this recovers Theorem 1.1. Now if we allow wild ramification, we can still continue with similar flavour. Let U be a smooth variety over k with char(k) 2. Choose a compactification U X with X normal and proper over k. We will define Chow group CH 0 (X, D) with modulus D where D is an cartier divisor with support D in X \ U. Then one can prove the following theorem, 5
Theorem 3.4. (Kerz-Saito) The reciprocity map induces an isomorphism CH 0 (X, D) π ab 1 (X, D) where π ab 1 (X, D) is the quotient of π ab 1 (X) which classifies abelian étale coverings of U with ramification over X \ U bounded by the divisor D. Remark 3.5. CH 0 (X, D) is extension of Chow group of zero cycles on U and also an extension of suslin homology. So we can recover the previous theorems. References [1] Szamuely, Tamás, Galois groups and fundamental groups, in Cambridge Studies in Advanced Mathematics., (2009). [2] Kato, Kazuya and Saito, Shuji, Unramified class field theory of arithmetical surfaces, in Ann. of Math., (1983), 241 275. [3] Lang, Serge, Unramified class field theory over function fields in several variables, in Ann. of Math., (1956), 285 325. 6