Arithmetic volumes for lattices over p-adic rings

In this paper we develop formulas for computing the arithmetic volume of orthogonal groups for lattices over the maximal orders of finite extensions of ZpZp. We specifically develop new explicit formulas for unimodular lattices over 2-adic rings. We also develop a reduction of the general problem to that of unimodular lattices together with the combinatorial problem of computing representatives for all possible Jordan decompositions.