On finite arithmetic groups

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:

algebraic integers , Galois groups , integral representations , realization fields