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

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 ${\mathbb{Q}}_2$. 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.