ON THE CLASSIFICATION OF THE MAXIMAL ARITHMETIC SUBGROUPS OF SIMPLY CONNECTED GROUPS

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.