Brauer Equivalence in a Homogeneous Space with Connected Stabilizer

Let G be a simply connected algebraic group over a field k of characteristic 0, H a connected k-subgroup of G, X = H\G. When k is a local field or a number field, we compute the set of Brauer equivalence classes in X(k). 0. Introduction In this note we investigate the Brauer equivalence in a homogeneous space X = H\G, where G is a simply connected algebraic group over a local field or a number field, and H is a connected subgroup of G. In more detail, let k be a field of characteristic 0, and let k̄ be a fixed algebraic closure of k. For a smooth algebraic variety Y over k, set Y = Yk̄ = Y ×k k̄. Let Br Y denote the cohomological Brauer group of Y , BrY = H ét (Y,Gm). Set Br1 Y = ker[BrY → BrY ]. There is a canonical pairing Y (k)× Br1 Y → Br k, (y, b) 7→ b(y) (0.1) called the Manin pairing. We define the Brauer equivalence on Y (k) as follows: y1 ∼ y2 if (y1, b) = (y2, b) for all b ∈ Br1 Y . We denote the set of classes of Brauer equivalence in Y (k) by Y (k)/Br . Note that we define the Brauer equivalence in terms of Br1 Y , not in terms of Br1 Y c or BrY , where Y c is a smooth compactification of Y . The notion of B-equivalence for a subgroup B of the Brauer group BrY was introduced by Manin [Ma1], [Ma2]. Colliot-Thélène and Sansuc [CT/Sa1] investigated the Brauer equivalence in algebraic tori (they defined the Brauer equivalence in terms of the Brauer group of a smooth This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities — Center of Excellence Program. The first named author was partially supported by the Hermann Minkowski Center for Geometry. The second named author was partially supported by the Ministry of Absorption (Israel) and the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. 1 2 MIKHAIL BOROVOI AND BORIS KUNYAVSKĬI compactification). The Brauer equivalence in reductive groups was studied in [Th]. Let G be a simply connected semisimple algebraic group over k. Let H be a connected subgroup of G. We denote by H tor the biggest toric quotient group of H . We are interested in the Brauer equivalence in the set X(k) where X = H\G. We compute X(k)/Br when k is a local field. Namely, we prove that there is a bijection X(k)/Br ∼ −→im [ker[H(k,H) → H(k,G)] → H(k,H )] (Theorem 2.1). Moreover, when k is a nonarchimedean local field, we prove that there is a bijection X(k)/Br → H(k,H ) (Theorem 2.2). We also compute X(k)/Br when k is a number field. We prove that there is a bijection