ON FINITE ARITHMETIC GROUPS

Let F be a finite extension of Q, Qp 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 GLn(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 2 G for some nite Galois stable subgroup G of GLn(C), there is only a finite number of possible realization field extensions of F if G  GLn(OE) over the ring OE 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.