Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part II, Expanded version

Abstract

This paper is the complementary work of [Cho16]. Ramified quadratic extensions $E/F$, where $F$ is a finite unramified field extension of $\mathbb{Q}_2$, fall into two cases that we call $\textit{Case 1}$ and $\textit{Case 2}$. In the previous work [Cho16], we obtained the local density formula for a ramified hermitian lattice in $\textit{Case 1}$. In this paper, we obtain the local density formula for the remaining $\textit{Case 2}$, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with the paper [GY00] of W. T. Gan and J.-K. Yu and [Cho16], allows the computation of the mass formula for any hermitian lattice $(L, H)$, when a base field is unramified over $\mathbb{Q}$ at a prime $(2)$.