Local density formulas of unimodular lattices over dyadic fields

Abstract

Recently Fiori obtained the local density formulas of unimodular lattices over dyadic local fields, but his results have contradictions with Pfeuffer's and Körner's results for rank 3 case, and in some cases his formulas of rank 4 have fractions in the exponent part which is not compatible with the definition of local density. In this paper, using Siegel's mass formula and neighbor lattice methods we obtain a relation between local density and the number of neighbor lattices of dyadic unimodular lattices. Using the relation we get the local densities of unimodular lattices of ranks 3 and 4 over dyadic local fields. Combining these with the results of Pfeuffer, we can obtain the local density formulas of general unimodular lattices over dyadic local fields. Keywords dyadic unimodular lattice, local density, Siegel's mass formula, neighbor lattice MSC(2010)11E08, 11E12