Prasad, Gopal [Jarden, Moshe] Volumes of \(S\)-arithmetic quotients of semi-simple groups. With an appendix by Moshe Jarden and Gopal Prasad. (English) Zbl 0695.22005 Publ. Math., Inst. Hautes Étud. Sci. 69, 91-117 (1989). Let \(k\) be a global field and \(G\) a semisimple algebraic group defined over \(k\). Let S be a finite set of places of \(k\) containing the archimedean ones. For \(G\) simply connected and absolutely quasi-simple a formula is given for the volume of \(G(k_S)/\Lambda\), where \(\Lambda\) is an \(S\)-arithmetic subgroup of \(G(k)\). The derivation of this formula is possible thanks to the theory of reductive groups over local fields of Bruhat and Tits. The same computations give a bound for class numbers. Reviewer: J. G. M. Mars (Utrecht) Cited in 7 ReviewsCited in 66 Documents MSC: 22E46 Semisimple Lie groups and their representations 20G30 Linear algebraic groups over global fields and their integers 22E40 Discrete subgroups of Lie groups 11R29 Class numbers, class groups, discriminants Keywords:global field; semisimple algebraic group; volume; S-arithmetic subgroup; reductive groups; bound for class numbers × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML