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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 6

# Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II

Sungmun Cho
• Corresponding author
• Graduate School of Mathematics, Kyoto University, Kitashirakawa, 606-8502, Kyoto, Japan; and POSTECH, Department of Mathematics, 7 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 37673, Pohang, Republic of Korea
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Published Online: 2018-08-07 | DOI: https://doi.org/10.1515/forum-2017-0080

## Abstract

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions $E/F$, where F is a finite unramified field extension of ${ℚ}_{2}$, fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice $\left(L,H\right)$, when a base field is unramified over $ℚ$ at a prime $\left(2\right)$.

MSC 2010: 11E41; 11E95; 14L15; 20G25; 11E39; 11E57

## References

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S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990. Google Scholar

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S. Cho, Group schemes and local densities of quadratic lattices in residue characteristic 2, Compos. Math. 151 (2015), no. 5, 793–827.

• [3]

S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 (2016), no. 3, 451–532.

• [4]

S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part II, preprint (2018), https://arxiv.org/abs/1806.00726.

• [5]

W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 (2000), no. 3, 497–524.

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R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. Google Scholar

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R. Jacobowitz, Hermitian forms over local fields, Amer. J. Math. 84 (1962), 441–465.

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C.-H. Sah, Quadratic forms over fields of characteristic 2, Amer. J. Math. 82 (1960), 812–830.

Published Online: 2018-08-07

Published in Print: 2018-11-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: KAKENHI Grant No. 16F16316

The author is partially supported by JSPS KAKENHI Grant No. 16F16316 and NRF 2018R1A4A 1023590.

Citation Information: Forum Mathematicum, Volume 30, Issue 6, Pages 1487–1520, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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