Volumes of \(S\)-arithmetic quotients of semi-simple groups. With an appendix by Moshe Jarden and Gopal Prasad. (English) Zbl 0695.22005

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.


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
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