On finite arithmetic groups

Let F be a finite extension of ℚ, 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 suffciently 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 ε G for some finite Galois stable subgroup G of GL_n(ℂ), there is only a finite number of possible realization field extensions of F if G ⊂ 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.