Group schemes and local densities of quadratic lattices in residue characteristic 2

Sungmun Cho, Purdue University

Abstract

The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of a quadratic lattice (L, Q) as a product of local factors, called the local densities of (L, Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme G over [special characters omitted] with the generic fiber [special characters omitted](L, Q), which satisfies G([special characters omitted](L, Q). Our method works for any unramified finite extension of [special characters omitted]. Therefore, we give the long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unrami fied finite extensions of [special characters omitted]. As an example, we give the mass formula for the integral quadratic form Qn(x1, ···, xn) = [special characters omitted] associated to a number field k which is totally real and such that the ideal (2) is unramified over k.

Degree

Ph.D.

Advisors

Yu, Purdue University.

Subject Area

Applied Mathematics|Mathematics

COinS