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On the classification of the maximal arithmetic subgroups of simply connected groups
A. A. Ryzhikov,
V. I. Chernousov Institute of Mathematics, National Academy of Sciences of the Republic of Belarus
Abstract:
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.
UDC:
512.743
MSC: Primary
20G15; Secondary
11E57,
14L35,
14L40 Received: 30.12.1996
DOI:
10.4213/sm260
Fulltext:
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