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

Cho, Sungmun

doi:10.2140/ant.2016.10.451

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

Sungmun Cho

Vol. 10 (2016), No. 3, 451–532

DOI: 10.2140/ant.2016.10.451

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 (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

Vol. 10, No. 3, 2016