Algebra & Number Theory

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

Sungmun Cho

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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.

Article information

Source
Algebra Number Theory, Volume 10, Number 3 (2016), 451-532.

Dates
Received: 30 August 2013
Revised: 15 September 2015
Accepted: 25 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842494

Digital Object Identifier
doi:10.2140/ant.2016.10.451

Mathematical Reviews number (MathSciNet)
MR3513129

Zentralblatt MATH identifier
1341.11016

Subjects
Primary: 11E41: Class numbers of quadratic and Hermitian forms
Secondary: 11E95: $p$-adic theory 14L15: Group schemes 20G25: Linear algebraic groups over local fields and their integers 11E39: Bilinear and Hermitian forms 11E57: Classical groups [See also 14Lxx, 20Gxx]

Keywords
local density mass formula group scheme smooth integral model

Citation

Cho, Sungmun. Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I. Algebra Number Theory 10 (2016), no. 3, 451--532. doi:10.2140/ant.2016.10.451. https://projecteuclid.org/euclid.ant/1510842494


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