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In this paper we give an explicit formula for the mass of a quadratic form in n f 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 [S]. §0. Summary Our goal is to give an exact formula for the mass of the genus of a quadratic form j 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 j 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 [S]. 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, j definite; v real, j indefinite; and v complex. To define the factors in the last two cas

Who reads An Exact Mass Formula for Quadratic Forms Over Number Fields?

Author

Jonathan Hanke

Publisher

Walter de Gruyter GmbH

Published

2005

Language

EN

Field

Mathematics (Physical Sciences)