Aa. Ryzhkov et Vi. Chernousov, ON THE CLASSIFICATION OF THE MAXIMAL ARITHMETIC SUBGROUPS OF SIMPLY CONNECTED GROUPS, Sbornik. Mathematics, 188(9-10), 1997, pp. 1385-1413
Let G subset of GL(n) be a simply connected simple algebraic group def
ined over a field K of algebraic numbers and let T be the set of all n
on-Archimedean valuations v of the field K. As is well known, each max
imal arithmetic subgroup Gamma subset of G can be uniquely recovered b
y means of some collection of parachoric subgroups; to be more precise
, there exist parachoric subgroups M-v subset of G(K-v), v is an eleme
nt of T, that have maximal types and satisfy the relation Gamma = N-G(
M), where M = G(K) boolean AND Pi(v is an element of T) M-v. Thus, the
re naturally arises the following question: for what collections {M-v}
(v is an element of T) of parachoric subgroups M-v subset of G(K-v) of
maximal types is the above subgroup Gamma subset of G a maximal arith
metic subgroup of G? Using Rohlfs's cohomology criterion for the maxim
ality of an arithmetic subgroup, necessary and sufficient conditions f
or the maximality of the above arithmetic subgroup Gamma subset of G a
re obtained. The answer is given in terms of the existence of elements
of the field K with prescribed divisibility properties.