Group schemes and local densities

2000 ◽  
Vol 105 (3) ◽  
pp. 497-524 ◽  
Author(s):  
Jiu-Kang Yu ◽  
Wee Teck Gan
Keyword(s):  
Group Schemes ◽  
Local Densities

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

2018 ◽  
Vol 30 (6) ◽  
pp. 1487-1520 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Mass Formula ◽  
Local Density ◽  
Group Scheme ◽  
Field Extension ◽  
Integral Group ◽  
Group Schemes ◽  
Quadratic Extensions ◽  
Local Densities ◽  
Characteristic 2 ◽  
Residue Characteristic

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 {\mathbb{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 {(L,H)} , when a base field is unramified over {\mathbb{Q}} at a prime {(2)} .

scholarly journals Group schemes and local densities of quadratic lattices in residue characteristic 2

2014 ◽  
Vol 151 (5) ◽  
pp. 793-827 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Mass Formula ◽  
Local Density ◽  
Finite Extension ◽  
Group Scheme ◽  
Generic Fiber ◽  
Group Schemes ◽  
Local Densities ◽  
Quadratic Lattices ◽  
Totally Real

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

scholarly journals Substrate wettability guided oriented self assembly of Janus particles

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Meneka Banik ◽  
Shaili Sett ◽  
Chirodeep Bakli ◽  
Arup Kumar Raychaudhuri ◽  
Suman Chakraborty ◽  
...  
Keyword(s):  
Self Assembly ◽  
Janus Particles ◽  
Water Molecules ◽  
Functional Coatings ◽  
Hexagonal Close Packed ◽  
Local Densities ◽  
Inhomogeneous Properties ◽  
Metal Polymer ◽  
Specific Orientation

AbstractSelf-assembly of Janus particles with spatial inhomogeneous properties is of fundamental importance in diverse areas of sciences and has been extensively observed as a favorably functionalized fluidic interface or in a dilute solution. Interestingly, the unique and non-trivial role of surface wettability on oriented self-assembly of Janus particles has remained largely unexplored. Here, the exclusive role of substrate wettability in directing the orientation of amphiphilic metal-polymer Bifacial spherical Janus particles, obtained by topo-selective metal deposition on colloidal Polymestyere (PS) particles, is explored by drop casting a dilute dispersion of the Janus colloids. While all particles orient with their polymeric (hydrophobic) and metallic (hydrophilic) sides facing upwards on hydrophilic and hydrophobic substrates respectively, they exhibit random orientation on a neutral substrate. The substrate wettability guided orientation of the Janus particles is captured using molecular dynamic simulation, which highlights that the arrangement of water molecules and their local densities near the substrate guide the specific orientation. Finally, it is shown that by spin coating it becomes possible to create a hexagonal close-packed array of the Janus colloids with specific orientation on differential wettability substrates. The results reported here open up new possibilities of substrate-wettability driven functional coatings of Janus particles, which has hitherto remained unexplored.

Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

2021 ◽  
pp. 1-38
Author(s):  
Yumiko Hironaka
Keyword(s):  
Laurent Polynomial ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Explicit Formulas ◽  
Hermitian Forms ◽  
Spherical Fourier Transform ◽  
Local Densities ◽  
Laurent Polynomial Ring ◽  
Injective Map ◽  
Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

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