#### Vol. 10, No. 3, 2016

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Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I

### Sungmun Cho

Vol. 10 (2016), No. 3, 451–532
##### Abstract

The obstruction to the local-global principle for a hermitian lattice $\left(L,H\right)$ can be quantified by computing the mass of $\left(L,H\right)$. The mass formula expresses the mass of $\left(L,H\right)$ as a product of local factors, called the local densities of $\left(L,H\right)$. The local density formula is known except in the case of a ramified hermitian lattice of residue characteristic 2.

Let $F$ be a finite unramified field extension of ${ℚ}_{2}$. Ramified quadratic extensions $E∕F$ fall into two cases that we call Case 1 and Case 2. In this paper, we obtain the local density formula for a ramified hermitian lattice in Case 1, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with the paper of W. T. Gan and J.-K. Yu (Duke Math. J. 105 (2000), 497–524), allows the computation of the mass formula for a hermitian lattice $\left(L,H\right)$ in Case 1.

##### Keywords
local density, mass formula, group scheme, smooth integral model
##### Mathematical Subject Classification 2010
Primary: 11E41
Secondary: 11E95, 14L15, 20G25, 11E39, 11E57