Implications of the Hasse Principle for Zero Cycles of Degree One on Principal Homogeneous Spaces

Let $k$ be a perfect field of virtual cohomological dimension $\leq 2$. Let $G$ be a connected linear algebraic group over $k$ such that $G^{sc}$ satisfies a Hasse principle over $k$. Let $X$ be a principal homogeneous space under $G$ over $k$. We show that if $X$ admits a zero cycle of degree one, then $X$ has a $k$-rational point.


Introduction
The following question of Serre [11, pg 192 ] is open in general.
Q: Let k be a field and G a connected linear algebraic group defined over k. Let X be a principal homogeneous space under G over k. Suppose X admits a zero cycle of degree one, does X have a k-rational point?
Let k be a number field, let V be the set of places of k and let k v denote the completion of k at a place v. We say that a connected linear algebraic group G defined over k satisfies a Hasse principle over k if the map H 1 (k, G) → v∈V H 1 (k v , G) is injective. Let V r denote the set of real places of k. If G is simply connected, then by a theorem of Kneser, the Hasse principle reduces to injectivity of the maps H 1 (k, G) → v∈Vr H 1 (k v , G). That this result holds is a theorem due to Kneser,Harder and Chernousov [4] , [5], [6]. Sansuc used this Hasse principle to show that Q has positive answer for number fields.
Let k be any field and Ω the set of orderings of k. For v ∈ Ω let k v denote the real closure of k at v. We say that a connected linear algebraic group G defined over k satisfies a Hasse principle over k if the map H 1 (k, G) → v∈Ω H 1 (k v , G) is injective. It is a conjecture of Colliot-Thélène [2, pg 652] that a simply connected semisimple group satisfies a Hasse principle over a perfect field of virtual cohomological dimension ≤ 2. Bayer and Parimala [2] have given a proof in the case where G is of classical type, type F 4 and type G 2 .
Our goal in this paper is to extend Sansuc's result by providing a positive answer to Q when k is a perfect field of virtual cohomological dimension ≤ 2 and G sc satisfies a Hasse principle over k. More precisely, we prove the following: Theorem 0.1. Let k be a perfect field of virtual cohomological dimension ≤ 2. Let {L i } 1≤i≤m be a set of finite field extensions of k such that the greatest common divisor of the degrees of the extensions [L i : k] is 1. Let G be a connected linear algebraic group over k. If G sc satisfies a Hasse principle over k, then the canonical map has trivial kernel.
We obtain the following as a corollary: Corollary 0.2. Let k be a perfect field of virtual cohomological dimension ≤ 2. Let {L i } 1≤i≤m be a set of finite field extensions of k such that the greatest common divisor of the degrees of the extensions [L i : k] is 1. Let G be a connected linear algebraic group over k. If the simple factors of G sc are of classical type, type F 4 or type G 2 then the canonical map Sansuc's proof of a positive answer to Q over number fields relies on the surjectivity of the map H 1 (k, µ) → v∈Vr H 1 (k v , µ) for µ a finite commutative group scheme. This result is a consequence of the Chebotarev density theorem and does not extend to a general field of virtual cohomological dimension ≤ 2. Even in the case µ = µ 2 , the surjectivity of the map H 1 (k, µ) → v∈Ω H 1 (k v , µ) imposes severe conditions on k like the SAP property. The main content of this paper is to replace the arithmetic in Sansuc's paper with a norm principle over a real closed field.

Algebraic Groups
In this section, we review some well-known facts from the theory of algebraic groups and define some notation used in the remainder of the work.
Let k be a field. An algebraic group G over k is a smooth group scheme of finite type. A surjective morphism of algebraic groups with finite kernel is called an isogeny of algebraic groups. An isogeny G 1 → G 2 is said to be central if its kernel is a central subgroup of G 1 .
An algebraic torus is an algebraic group T such that T (k) is isomorphic to a product of multiplicative groups G m,k . A torus T is said to be quasitrivial if it is a product of groups of the form R Ei/k G m where {E i } 1≤i≤r is a family of finite field extensions of k.
An algebraic group G is called linear if it is isomorphic to a closed subgroup of GL n form some n, or equivalently, if its underlying algebraic variety is affine. Of particular interest among connected linear algebraic groups are semisimple groups and reductive groups.
A connected linear algebraic group is called semisimple if it has no nontrivial, connected, solvable, normal subgroups. A semisimple group G is said to be simply connected if every central isogeny G ′ → G is an isomorphism. We can associate to any semisimple group a simply connected groupG (unique up to isomorphism) such that there is a central isogenyG → G. We refer toG as the simply connected cover of G.
Any simply connected semisimple group is a product of simply connected simple algebraic groups [7,Theorem 26.8]. Any simple algebraic group belongs to one of four infinite families A n , B n , C n , D n or is of type E 6 , E 7 , E 8 , F 4 or G 2 (see for example [7, §26]). A simple group which is of type A n , B n , C n or D n but not of type trialitarian D 4 is said to be a classical group. All other simple groups are called exceptional groups.
A connected linear algebraic group is called reductive if it has no nontrivial, connected, unipotent, normal subgroups. Given a connected linear algebraic group G, the unipotent radical of G denoted G u is the maximal connected unipotent normal subgroup of G. It is clear that G/G u is always a reductive group. We denote G/G u by G red . The commutator subgroup of G red is a semisimple group which we denote G ss . We denote the simply connected cover of G ss by G sc .
A special covering of a reductive group G is an isogeny where G 0 is a simply connected semisimple algebraic k-group and S is a quasitrival k-torus. Given a reductive group G there exists an integer n and a quasitrival torus T such that G n × T admits a special covering [9, Lemme 1.10].

Galois Cohomology and Zero Cycles
For our convenience, we will discuss Q in the context of Galois Cohomology. We briefly review some of the notions from Galois Cohomology we will use and then restate Q in this setting.
Let k be a field and Γ k = Gal(k/k) be the absolute Galois group of k. For an algebraic k-group G, let H i (k, G) = H i (Γ k , G(k)) denote the Galois Cohomology of G with the assumption i ≤ 1 if G is not abelian. For any k-group G, H 0 (k, G) = G(k) and H 1 (k, G) is a pointed set which classifies the isomorphism classes of principal homogeneous spaces under G over k. The point in H 1 (k, G) corresponds to the principal homogeneous space with rational point. We will interchangeably denote the point in H 1 (k, G) by point or 1.
Each Γ k -homomorphism f : G → G ′ induces a functorial map H i (k, G) → H i (k, G ′ ) which we shall also denote by f . Given an exact sequence of k-groups, there exists a connecting map δ 0 : G 3 (k) → H 1 (k, G 1 ) such that the following is an exact sequence of pointed sets.
there is in addition a connecting map δ 1 : H 1 (k, G 3 ) → H 2 (k, G 1 ) such that the following is an exact sequence of pointed sets.
Given a field extension L of k, Gal(k/L) ⊂ Gal(k/k) and there is a restriction homomorphism res : H 1 (k, G) → H 1 (L, G). If G is a commutative group, and if the degree of L over k is finite, there is also a corestriction homomorphism cores : H 1 (L, G) → H 1 (k, G). The composition cores • res is multiplication by the degree of L over k.
Let p be any prime number. The p-cohomological dimension of k is less than or equal to r (written cd p (k) ≤ r) if H n (k, A) = 0 for every p-primary torsion Γ kmodule A and n > r. The cohomological dimension of k is less than or equal to r (written cd(k) ≤ r), if cd p (k) ≤ r for all primes p. Finally, the virtual cohomological dimension of k, written vcd(k) is precisely the cohomological dimension of k( √ −1). If k is a field of positive characteristic then vcd(k) = cd(k).
Let X be a scheme. For any closed point x ∈ X, let O x be the local ring at x and let M x be its maximal ideal. The residue field of x written k( Zero cycles of X are elements of the free abelian group on closed points x ∈ X. We may associate to any zero cycle A closed point with residue field k is called a rational point. It is clear that if x is a closed point of a variety X over k then it is a rational point of X k(x) . We have seen that the point in H 1 ( * , G) is the principal homogeneous space under G over * with a rational point. Therefore, a principal homogeneous space X under G over k, with zero cycle n i x i is an element of the kernel of the product of the restriction maps H 1 (k, G) → H 1 (k(x i ), G). If the zero cycle is of degree one, then the field extensions k(x i ) are necessarily of coprime degree over k.
Guided by this insight, one may restate Q as follows.
Q: Let k be a field and let G be a connected, linear algebraic group defined over k. Let {L i } 1≤i≤m be a collection of finite extensions of k with gcd([L i : k]) = 1. Does the canonical map have trivial kernel?

Orderings of a Field
We recall some basic properties of orderings of a field [10]. An ordering ν of a field k is given by a binary relation ≤ ν such that for all a, b, c ∈ k. • and 0 ≤ v c then ca ≤ v cb A field k which admits an ordering is necessarily of characteristic 0. If k is a field with an ordering v, an algebraic extension of the ordered field (k, v) is an algebraic field extension L of k together with an ordering v ′ on L such that v ′ restricted to k is v. If L is a finite field extension of k of odd degree there is always an algebraic extension (L, v ′ ) of (k, v) [10, Chapter 3, Theorem 1.10].
A field k is said to be formally real if -1 is not a sum of squares in k. A field k is called a real closed field if it is a formally real field and no proper algebraic extension is formally real. There is a unique ordering on a real closed field. This ordering is defined by the relation a ≤ b if and only if b − a is a square in k. Further, if k is a real closed field, then k( √ −1) is algebraically closed [10, Theorem 2. 3 (iii)]. If L is a finite field extension of k, then k v ⊗ L is isomorphic to a product of the form is an algebraic closure for k there is a natural inclusion Gal(k, k v ) ⊂ Γ k and thus a restriction map H 1 (k, G) → H 1 (k v , G).

Main Result
In the discussion which follows we will need the following lemmas.
Lemma 4.1. Let k be a field and let G be a reductive group over k. Fix an integer n and a quasitrivial torus T such that G n × T admits a special covering Then G sc satisfies a Hasse Principle over k if and only if G 0 satisfies a Hasse principle over k.
Proof. Taking commutator subgroups we have a short exact sequence which in turn is (G ss ) n by definition of G ss . Therefore, we have the following short exact sequence 1 →μ → G 0 → (G ss ) n → 1 whereμ is some finite group schme. In particular, G 0 is a simply connected cover of (G ss ) n . Since (G sc ) n is certainly a simply connected cover of (G ss ) n , uniqueness of the simply connected cover of (G ss ) n gives (G sc ) n ∼ = G 0 . In particular, the simple factors of G sc are the same as the simple factors of G 0 and G sc satisfies the Hasse principle over k if and only if G 0 satisfies the Hasse principle over k.
Lemma 4.2. Let k be a real closed field and let G be a reductive group over k which admits a special covering Let L be a finiteètale k-algebra. Let δ be the first connecting map in Galois Cohomology and let N L/k denote the corestriction map H 1 (k ⊗ L, µ) → H 1 (k, µ). Then Proof. Since k is real closed, there exists finite numbers r and s such that k ⊗ L is isomorphic to a product of r copies of k and s copies of k( √ −1). Thus Since k is real closed, k( √ −1) is algebraically closed, H 1 (k( √ −1), µ) is trivial and H 1 (k ⊗ L, µ) is just a product of r copies of H 1 (k, µ). Therefore, is just the product map r copies That k ⊗ L is a product of r copies of k and s copies of k( √ −1) also gives that Therefore, the connecting map r copies is just the product of the connecting maps and the latter of which is necessarily the trivial map. So choose (x 1 , . . . , x r , y 1 , . . . , y s ) ∈ G(k ⊗ L) Then N L/k (δ(x 1 , . . . , x r , y 1 , . . . , y s )) = N L/k (δ(x 1 ), . . . , δ(x r ), δ(y 1 ), . . . , δ(y s )) = δ(x 1 ) · · · δ(x r ) = δ(x 1 · · · x r ) Since the x i were chosen to be in G(k) for all i, then x 1 · · · x r ∈ G(k) and the desired result holds. Proof. By [10, Chapter 3, Theorem 1.10] each ordering v of k extends to an ordering w of L, in particular each real closure k v is L w for some ordering w on L. Since the natural map H 1 (k, G) → H 1 (L w , G) factors through the canonical map H 1 (k, G) → H 1 (L, G), the desired result is immediate.
We now return to the result which is the main goal of this paper.
Theorem 4.4. Let k be a perfect field of virtual cohomological dimension ≤ 2 and let G be a connected linear algebraic group over k. Let {L i } 1≤i≤m be a set of finite field extensions of k such that the greatest common divisor of the degrees of the extensions [L i : k] is 1. If G sc satisfies a Hasse principle over k, then the canonical map has trivial kernel.
Applying [2, Theorem 10.1] a Serre twist we obtain the following corollary: Corollary 4.5. Let k be a perfect field of virtual cohomological dimension ≤ 2.
Let {L i } 1≤i≤m be a set of finite field extensions of k such that the greatest common divisor of the degrees of the extensions [L i : k] is 1. Let G be a connected linear algebraic group over k. If the simple factors of G sc are of classical type, type F 4 or type G 2 then the canonical map is injective.