On almost strong approximation for algebraic groups

If G is a simply connected reductive group defined over a number field k and ∞ is the set of all infinite places of k, then G has strong approximation with respect to ∞ if and only if the archimedean part of any k-simple component of the adèle group is non-compact. Using affine Bruhat–Tits buildings we formulate an almost strong approximation (ASAP) for groups of compact type, extending the version treated in [J.S. Hsia, M. Jöchner, Invent. Math. 129 (1997) 471–487]. The validity of ASAP for G(k) is proved for all classical groups of compact type whose Tits indices over k are not 2An(d) with d 3. Application to genera of integral forms (similar to Grossʹ notion of -models [B. Gross, Invent. Math. 124 (1996) 263–279]) is given with attendant results on integral representations of positive definite quadratic, hermitian or skew-hermitian forms.