Products of transvections in one conjugacy class of the symplectic group over the p-adic numbers

Abstract

Every element in the symplectic group over the field of p-adic numbers ( p>3) is a product of transvections in a single conjugacy class. We determine the minimal number of factors needed in any such product for transformations with path dimensions 1, 2, and 3. For indecomposable symplectic transformations with path dimensions 4, 5, and 6 we find upper bounds for the minimal number of factors. Results of Knüppel can now be applied to obtain similar upper bounds for transformations with higher path dimensions.