An exact mass formula for quadratic forms over number fields

Abstract

In this paper we give an explicit formula for the mass of a quadratic form in n ≧ 3 variables with respect to a maximal lattice over an arbitrary number field k , and use this to find the mass of many a-maximal lattices. We make the minor technical assumption that locally the determinant of the form is a unit up to a square if n is odd. The corresponding formula for k totally real was recently computed by Shimura [ G. Shimura , An exact mass formula for orthogonal groups, Duke Math. J. 97 (1999), no. 1, 1–66].