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

TL;DR

This paper derives a local density formula for ramified hermitian lattices in residue characteristic 2, specifically in Case 1, by constructing a smooth integral model, advancing the understanding of mass formulas in this context.

Contribution

It provides the first explicit local density formula for ramified hermitian lattices in residue characteristic 2 in Case 1, using a new integral group scheme model.

Findings

Derived local density formula for Case 1 ramified hermitian lattices.

Constructed a smooth integral group scheme model for the unitary group.

Enabled computation of the mass formula for these lattices.

Abstract

The obstruction to the local-global principle for a hermitian lattice (L, H) can be quantified by computing the mass of (L, H). The mass formula expresses the mass of (L, H) as a product of local factors, called the local densities of (L, H). 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, allows the computation of the mass formula for a hermitian lattice (L, H) in Case 1.