The Structure of the Symplectic Group Over a Valuation Ring

Abstract

More recently Klingenberg [6] effected a significant generalization of these results by extending them to arbitrary commutative local rings (with suitable residue class field). In doing so however, he assumed that the underlying alternating form is unimodular, i. e. has unit discriminant. He was able to prove in this case that every normal subgroup of the symplectic group is a congruence subgroup in the usual sense. In this paper we shall generalize these latter results by dropping the requirement that the discriminant be a unit, and requiring only that it be non-zero (i. e. that the alternating form be non-degenerate). To do this it is necessary to restrict the ring to be a valuation ring (with residue class field #L F3 and not of characteristic 2, the same assumptions made by Klingenberg). In this situation it turns out that there are normal subgrotps which are not congruence subgroups. In order to regain a complete description of the normal subgroups, we generalize the concept of congruence subgroup by allowing the congruence ideal to vary from entry to entry in the miatrix. (A precise definition is