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

Part of: Forms and linear algebraic groups Linear algebraic groups and related topics Algebraic groups

Published online by Cambridge University Press:  05 December 2014

Sungmun Cho
Sungmun Cho*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA email [email protected] Current address: Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
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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 $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

Keywords

local densitiesmass formulagroup schemessmooth integral models

MSC classification

Primary: 11E41: Class numbers of quadratic and Hermitian forms 11E95: $p$-adic theory 14L15: Group schemes 20G25: Linear algebraic groups over local fields and their integers
Secondary: 11E12: Quadratic forms over global rings and fields 11E57: Classical groups
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 5 , May 2015 , pp. 793 - 827
Copyright
© The Author 2014 

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