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On almost strong approximation for algebraic groups

✍ Scribed by Wai Kiu Chan; J.S. Hsia

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 200 KB
Volume: 254
Category: Article
ISSN: 0021-8693
DOI: 10.1016/s0021-8693(02)00100-x

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✦ Synopsis

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 G A 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 2 A (d) n with d 3. Application to genera of integral forms (similar to Gross' notion of Z-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.

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