On finite arithmetic groups

Document Type: Ischia Group Theory 2012

Author

I.H.E.S.

Abstract

Let $F$ be a finite extension of $\Bbb Q$‎, ‎${\Bbb Q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $E/F$ be a Galois extension‎. ‎We study the realization fields of‎ ‎finite subgroups $G$ of $GL_n(E)$ stable under the natural‎ ‎operation of the Galois group of $E/F$‎. ‎Though for sufficiently large $n$ and a fixed‎ ‎algebraic number field $F$ every its finite extension $E$ is‎ ‎realizable via adjoining to $F$ the entries of all‎ ‎matrices $g\in G$ for some finite Galois stable subgroup $G$ of $GL_n(\Bbb C)$‎, ‎there is only a‎ ‎finite number of possible realization field extensions of $F$ if $G\subset GL_n(O_E)$ over the‎ ‎ring $O_E$ of integers of $E$‎. ‎After an exposition of earlier results we give their refinements‎ ‎for the‎ ‎realization fields $E/F$‎. ‎We consider some applications to quadratic lattices‎, ‎arithmetic algebraic geometry and Galois cohomology of related arithmetic groups‎.

Keywords

Main Subjects


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