Vol. 10, No. 3, 2016

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I

Sungmun Cho

Vol. 10 (2016), No. 3, 451–532
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 2. Ramified quadratic extensions EF 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 (L,H) in Case 1.

Keywords
local density, mass formula, group scheme, smooth integral model
Mathematical Subject Classification 2010
Primary: 11E41
Secondary: 11E95, 14L15, 20G25, 11E39, 11E57
Milestones
Received: 30 August 2013
Revised: 15 September 2015
Accepted: 25 October 2015
Published: 12 June 2016
Authors
Sungmun Cho
Department of Mathematics
University of Toronto
40 St. George St., Room 6290
Toronto ON M5S 2E4
Canada