An exact mass formula for quadratic forms over number fields
Journal für die Reine und Angewandte Mathematik
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 -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 [Shi]. §0 Summary Our goal is to give an exact formula for the mass of the genus of
... ss of the genus of a quadratic form ϕ on a maximal lattice defined over an arbitrary number field k. In §2 we explain how knowledge of the Tamagawa number of the special orthogonal group G ϕ gives rise to a mass formula. Such a formula expresses the mass as a product of local factors over all places v of k, so our problem is reduced to computing each of these. For the non-archimedian places, these factors were recently computed by Shimura [Shi]. We state his result in §3 and for completeness include a translation between our language and his. In §4 we compute the archimedian factors, treating separately the 3 cases: v real, ϕ definite; v real, ϕ indefinite; and v complex. To define the factors in the last two cases, we choose a symmetric space Z v on which G ϕ v acts and a non-zero G ϕ v -invariant volume form ω Z . Finally, in §5 we compute the mass of ϕ with respect to a maximal lattice. We note that this formula agrees with Shimura's when k is totally real. In §6 we conclude by using the local similitude groups to show that this agrees with the mass of many genera of a-maximal lattices. Our results depend on several technical lemmas which we include as an appendix.