On weak approximation in homogeneous spaces of simply connected algebraic groups. (English) Zbl 0787.14009

Automorphic functions and their applications, Int. Conf., Khabarovsk/USSR 1988, 64-81 (1990).
[For the entire collection see Zbl 0727.00003.]
Let \(K\) denote an algebraic number field. Let \(S\) be a finite set of its places. An algebraic variety \(X\) over \(K\) satisfies the condition of weak approximation (\(\text{WA}_ S\)) with respect to \(S\), provided \(X(K)\), the set of \(K\)-rational points of \(X\), is dense in \(\prod_{v\in S} X(K_ v)\), where \(K_ v\) denotes the completion of \(K\) at \(v\). – Let \(T\) be an algebraic torus. The finite abelian group \(A_ S(T) = (\prod_{v\in S}T(K_ v))/T(K)_ S\), where \(T(K)_ S\) denotes the closure of \(\prod_{v\in S} T(K_ v)\), measures the failure of \((\text{WA}_ S\)). In particular (\(\text{WA}_ S\)) holds if and only if \(A_ S(T) = 0\). The group \(A_ S(T)\) has been studied by V. E. Voskresenskij [cf. Math. USSR, Izv. 4(1970), 1-17 (1971); translation from Izv. Akad. Nauk SSSR 34, 3-19 (1970; Zbl 0254.14016), and “Algebraic tori” (Russian; Moscow 1977; Zbl 0499.14013)] and by J.-J. Sansuc [J. Reine Angew. Math. 327, 12-80 (1981; Zbl 0468.14007)].
In the present paper the author studies the case of \(X = H\setminus G\), where \(H\) is a connected \(K\)-subgroup of a simply connected algebraic \(K\)-group \(G\). The main results concern an abelian group, called ‘defect’ of (\(\text{WA}_ S\)) for \(X\), depending only on \(H\), in particular its computation as a Brauer group. Proofs are based on a generalization of the duality theory of Tate and Nakayama from the case of tori to connected reductive \(K\)-groups done by R. E. Kottwitz [Duke Math. J. 51, 611-650 (1984; Zbl 0576.22020) and Math. Ann. 275, 365-399 (1986; Zbl 0577.10028)]. – For the Hasse principle of homogeneous spaces see also the author in J. Reine Angew. Math. 426, 179-192 (1992; Zbl 0739.14030).
Reviewer: P.Schenzel (Halle)


14G05 Rational points
14M17 Homogeneous spaces and generalizations
14B12 Local deformation theory, Artin approximation, etc.
11R34 Galois cohomology
13B40 √Čtale and flat extensions; Henselization; Artin approximation