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 ≤ 2. Let G be a connected linear algebraic group over k such that Gsc 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 [10, p. 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 kv 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(k,G) → ∏ v∈V H (kv, G) is injective. Let Vr 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(k,G) → ∏ v∈Vr H (kv, G). That this result holds is a theorem due to Kneser, Harder and Chernousov [3], [4], [5]. Sansuc used this Hasse principle to show that Q has a positive answer for number fields. Let k be any field and Ω the set of orderings of k. For v ∈ Ω let kv 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(k,G) → ∏ v∈Ω H (kv, G) is injective. It is a conjecture of Colliot-Thélène [1, p. 652] that a simply connected semisimple group satisfies a Hasse principle over a perfect field of virtual cohomological dimension ≤ 2. Bayer and Parimala [1] have given a proof in the case where G is of classical type, type F4 and type G2. 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 satisfies a Hasse principle over k. More precisely, we prove the following: Received by the editors October 7, 2010. 2010 Mathematics Subject Classification. Primary 11E72; Secondary 11E57. The results in this work are from a doctoral dissertation in progress under the direction of R. Parimala, whom the author sincerely thanks for her guidance. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication