On almost strong approximation for some exceptional groups.
The main object of the paper under review is a simply connected semisimple linear algebraic group \(G\) defined over a number field \(k\). \(G\) satisfies the strong approximation property with respect to a finite set \(S\) of places of \(k\) if and only if \(G_S=\prod_{{\mathfrak p}\in S}G(k_{\mathfrak p})\) is not compact. Thus if, say, \(G_\infty\) is not compact, one has to use a place \(\mathfrak q\) such that strong approximation holds for \(G\) with respect to \(\{\mathfrak q\}\cup\infty\). In a situation like that, inspired by the paper of \textit{J.~S.~Hsia} and \textit{M.~Jöchner} [Invent. Math. 129, No. 3, 471-487 (1997; Zbl 0894.11013)], where the case of quadratic forms was treated, \textit{W.~K.~Chan} and \textit{J.~S.~Hsia} [J. Algebra 254, No. 2, 441-461 (2002; Zbl 1027.20024)] introduced the notion of almost strong approximation property (ASAP) which allows one to partially recover the loss of the arithmetical information due to throwing away the place \(\mathfrak q\), proved ASAP for all classical groups of compact type whose Tits index over \(k\) is not \(^2A^{(d)}_n\) (\(d\geq 3\)), and applied this property to the study of integral forms of \(G\). In the paper under review, ASAP is established for all groups of compact type defined over a totally real number field \(k\) for the following types: \(^{3,6}D_4\), \(G_2\), \(F_4\), \(E_7\), and \(E_8\). Note that the proof is not case-by-case, as for classical groups, but relies upon the fact that all the above groups are quasi-split over a quadratic extension of \(k\).
simply connected semisimple groups over number fields
scientific article; zbMATH DE number 2102467
scientific article