An exact mass formula for quadratic forms over number fields

Jonathan Hanke
2005 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
more » ... 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.
doi:10.1515/crll.2005.2005.584.1 fatcat:wo4y2ykzzzfjzoedtqcxrzmogm