On the classification of the maximal arithmetic subgroups of simply connected groups

Abstract

Let be a simply connected simple algebraic group defined over a field of algebraic numbers and let be the set of all non-Archimedean valuations of the field . As is well known, each maximal arithmetic subgroup can be uniquely recovered by means of some collection of parachoric subgroups; to be more precise, there exist parachoric subgroups , , that have maximal types and satisfy the relation , where . Thus, there naturally arises the following question: for what collections of parachoric subgroups of maximal types is the above subgroup a maximal arithmetic subgroup of ? Using Rohlfs's cohomology criterion for the maximality of an arithmetic subgroup, necessary and sufficient conditions for the maximality of the above arithmetic subgroup are obtained. The answer is given in terms of the existence of elements of the field with prescribed divisibility properties.