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

2016 ◽  
Vol 10 (3) ◽  
pp. 451-532 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Group Schemes ◽  
Local Densities ◽  
Characteristic 2 ◽  
Residue Characteristic

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 Cohomological Invariants in Positive Characteristic

2021 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Positive Characteristic ◽  
Differential Forms ◽  
Symmetric Groups ◽  
Characteristic P ◽  
Motivic Cohomology ◽  
Cohomological Invariants ◽  
Group Schemes ◽  
Characteristic 2 ◽  
Theory Of Fields ◽  
Mod P

Abstract We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are invariants of $G$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $p$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $p$ Milnor K-theory of fields.

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 Semistable models of elliptic curves over residue characteristic 2

2020 ◽  
pp. 1-9
Author(s):  
Jeffrey Yelton
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Mixed Characteristic ◽  
Semistable Model ◽  
Characteristic 2 ◽  
Residue Characteristic ◽  
Legendre Form

Abstract Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$ , we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $ . We provide several examples along with an easy corollary concerning $2$ -torsion.

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 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.

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

scholarly journals Germline Transformation of Drosophila virilis With the Transposable Element mariner

Genetics ◽  
1996 ◽  
Vol 143 (1) ◽  
pp. 365-374 ◽  
Author(s):  
Allan R Lohe ◽  
Daniel L Hartl
Keyword(s):  
Transposable Element ◽  
Common Ancestor ◽  
Pcr Amplification ◽  
Drosophila Virilis ◽  
Cell Region ◽  
Diverse Species ◽  
Drosophila Mauritiana ◽  
Characteristic 2 ◽  
In Situ Hybridizations

Abstract An important goal in molecular genetics has been to identify a transposable element that might serve as an efficient transformation vector in diverse species of insects. The transposable element mariner occurs naturally in a wide variety of insects. Although virtually all mariner elements are nonfunctional, the Mosl element isolated from Drosophila mauritiana is functional. Mosl was injected into the pole-cell region of embryos of D. virilis, which last shared a common ancestor with D. mauritiana 40 million years ago. Mosl PCR fragments were detected in several pools of DNA from progeny of injected animals, and backcross lines were established. Because Go lines were pooled, possibly only one transformation event was actually obtained, yielding a minimum frequency of 4%. Mosl segregated in a Mendelian fashion, demonstrating chromosomal integration. The copy number increased by spontaneous mobilization. In situ hybridization confirmed multiple polymorphic locations of Mosl. Integration results in a characteristic 2-bp TA duplication. One Mosl element integrated into a tandem array of 370-bp repeats. Some copies may have integrated into heterochromatin, as evidenced by their ability to support PCR amplification despite absence of a signal in Southern and in situ hybridizations.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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