Some local-global principles in the arithmetic of algebraic groups over real function fields
In his seminal papers [J. Reine Angew. Math. 171, 4-11 (1934; Zbl 0009.29103); ibid. 176, 31-44 (1936; Zbl 0015.05701)] \textit{E. Witt} was the first to explore the arithmetic of algebraic groups over a real function field in one variable, by establishing first the local-global principle (which subsequently is called also the Hasse principle) for Brauer groups and then for quadratic forms over such fields. These two results are of fundamental importance in the arithmetic of quadratic and hermitian forms and related algebraic groups over these fields. It is natural and important to study, continuing the study of Witt et al., the arithmetic of algebraic groups over function fields in general, and over real function fields in particular, due to their connections with problems in algebraic geometry, algebraic \(K\)-theory and theory of quadratic and hermitian forms over rings. Let \(G\) be a semisimple group defined over the field \(k = \mathbb{R}(t)\), \(V\) the set of all valuations of \(k\) trivial on \(\mathbb{R}\). Let \(C(A)\) denote the center of a group \(A\). We say that \(A\) is projectively simple if \(A/C(A)\) is simple as abstract group. In this paper we prove that if \(G\) is \(k\)-isotropic and simply connected then \(G\) satisfies weak approximation with respect to \(V\). We prove a Hasse principle for isotropicity of almost simple \(k\)-groups of type different from \(^6D_4\), i.e. an almost simple \(k\)-group \(G\) of type \(\neq{^6D_4}\) is isotropic if and only if it is over all completions \(k_v\), \(v \in V\), of \(k\). A vertex of a Dynkin diagram of an almost simple group defined over a field \(K\) is called (after Tits) isotropic (resp. quasi-isotropic) if it is circled (resp. if there is another vertex in its orbit under the \(*\)-action of \(\text{Gal}(\overline K/K)\)). We show that a vertex of Dynkin diagram of an almost simple \(k\)-group \(G\) of type different from \(^3 D_4\) (resp. \(^3 D_4\), \(^6 D_4\)) is isotropic (resp. quasi-isotropic) if and only if it is so over all completions \(k_v\), \(v \in V\), of \(k\). As consequence, we derive the following important information about the structure of projective homogeneous spaces of semisimple \(k\)-groups and the normal structure of rational points of simply connected \(k\)-groups. 1) Any projective homogeneous space \(X\) over \(k\) of a semisimple \(k\)-group \(G\) without trialitarian factors satisfies the Hasse principle, i.e., if \(X(k_v) \neq \emptyset\) \(\forall v \in V\) then \(X(k) \neq \emptyset\); 2) Let \(G\) be an almost simple simply connected \(k\)-group not of type \(^6 D_4\). The following statements are equivalent: a) \(G\) is \(k\)-isotropic. b) \(G\) is \(k_v\)-isotropic for all \(v \in V\). c) \(G(k)\) is projectively simple and dense in \(G(k_v)\) for all \(v \in V\). d) \(G(k_v)\) is projectively simple for all \(v \in V\). Finally we prove a general theorem about the validity of cohomological Hasse principle (for \(H^1\)) of semisimple \(k\)-groups. In particular, if \(G\) is a semisimple \(\mathbb{R}\)-group, then considered as \(k\)-group, \(G\) satisfies cohomological Hasse principle, i.e., the natural nap \[ H^1(k,G) \to \prod_{v \in V} H^1(k_v,G) \] has trivial kernel (in the sense of pointed sets).
scientific article; zbMATH DE number 861930
scientific article