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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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