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 $\text{Case 1}$ and $\text{Case 2}$. In the previous work [Cho16], we obtained the local density formula for a ramified hermitian lattice in $\text{Case 1}$. In this paper, we obtain the local density formula for the remaining $\text{Case 2}$, 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 $(2)$.

89 pages. This is the expanded version of the published one. One error in the last line of Appendix B of 'Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I, Algebra & Number Theory, 10-3, 451-532, 2016' is explained in Remark B.1 of this version