1 results
Group schemes and local densities of quadratic lattices in residue characteristic 2
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 5 / May 2015
- Published online by Cambridge University Press:
- 05 December 2014, pp. 793-827
- Print publication:
- May 2015
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- Article
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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 
$k$.
