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 Mat. Sb., 1997, Volume 188, Number 9, Pages 127–156 (Mi msb260)

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.

DOI: https://doi.org/10.4213/sm260

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English version:
Sbornik: Mathematics, 1997, 188:9, 1385–1413

Bibliographic databases:

UDC: 512.743
MSC: Primary 20G15; Secondary 11E57, 14L35, 14L40

Citation: A. A. Ryzhikov, V. I. Chernousov, “On the classification of the maximal arithmetic subgroups of simply connected groups”, Mat. Sb., 188:9 (1997), 127–156; Sb. Math., 188:9 (1997), 1385–1413

Citation in format AMSBIB
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