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

A. A. Ryzhikov ; V. I. Chernousov

Sbornik. Mathematics, Tome 188 (1997) no. 9, pp. 1385-1413

Résumé

Let $G\subset \operatorname {GL}_n$ be a simply connected simple algebraic group defined over a field $K$ of algebraic numbers and let $T$ be the set of all non-Archimedean valuations $v$ of the field $K$. As is well known, each maximal arithmetic subgroup $\Gamma \subset G$ can be uniquely recovered by means of some collection of parachoric subgroups; to be more precise, there exist parachoric subgroups $M_v\subset G(K_v)$, $v\in T$, that have maximal types and satisfy the relation $\Gamma ={\mathrm N}_G(M)$, where $M=G(K)\cap \prod _{v\in T}M_v$. Thus, there naturally arises the following question: for what collections $\{M_v\}_{v\in T}$ of parachoric subgroups $M_v\subset G(K_v)$ of maximal types is the above subgroup $\Gamma \subset G$ a maximal arithmetic subgroup of $G$? Using Rohlfs's cohomology criterion for the maximality of an arithmetic subgroup, necessary and sufficient conditions for the maximality of the above arithmetic subgroup $\Gamma \subset G$ are obtained. The answer is given in terms of the existence of elements of the field $K$ with prescribed divisibility properties.