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On almost strong approximation for algebraic groups. (English) Zbl 1027.20024
If \(G\) is a simply connected reductive group defined over a number field \(k\) and \(\infty\) is the set of all infinite places of \(k\), then \(G\) has strong approximation with respect to \(\infty\) if and only if the Archimedean part of any \(k\)-simple component of the adèle group \(G_\mathbb{A}\) is non-compact. Using affine Bruhat-Tits buildings the authors formulate an almost strong approximation (ASAP) for groups of compact type. The validity of ASAP for \(G(k)\) is proved for all classical groups of compact type whose Tits indices over \(k\) are not \(^2A_n^{(d)}\) with \(d\geq 3\). Some consequences of ASAP in the study of the genera of integral forms of algebraic groups of compact type in the sense of B. H. Gross [Invent. Math. 124, No. 1-3, 263-279 (1996; Zbl 0846.20049)] are presented. When specialized to the groups SU and Spin for quadratic or skew-quadratic forms, the authors obtain results on integral representation of positive definite quadratic and Hermitian or skew-Hermitian forms.

MSC:
20G30 Linear algebraic groups over global fields and their integers
11E12 Quadratic forms over global rings and fields
11E57 Classical groups
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