On finite arithmetic groups

Abstract

Let $F$ be a finite extension of $Bbb Q$, ${Bbb Q}_p$ or a globalfield of positive characteristic, and let $E/F$ be a Galois extension.We study the realization fields offinite subgroups $G$ of $GL_n(E)$ stable under the naturaloperation of the Galois group of $E/F$. Though for sufficiently large $n$ and a fixedalgebraic number field $F$ every its finite extension $E$ isrealizable via adjoining to $F$ the entries of allmatrices $gin G$ for some finite Galois stable subgroup $G$ of $GL_n(Bbb C)$, there is only afinite number of possible realization field extensions of $F$ if $Gsubset GL_n(O_E)$ over thering $O_E$ of integers of $E$. After an exposition of earlier results we give their refinementsfor therealization fields $E/F$. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.

Keywords

• algebraic integers
• Galois groups
• integral representations
• realization fields