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

TL;DR

This paper completes the local density formulas for ramified hermitian lattices over quadratic extensions in residue characteristic 2, enabling comprehensive mass formula computations for such lattices.

Contribution

It provides the local density formula for Case 2 of ramified hermitian lattices by constructing a smooth integral group scheme model, complementing previous work on Case 1.

Findings

Derived local density formula for Case 2 lattices

Constructed smooth integral group scheme model for unitary groups

Enabled mass formula calculations for hermitian lattices over unramified fields at 2

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 $(2)$.