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 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 A note on local densities of quadratic forms

1983 ◽  
Vol 92 ◽  
pp. 145-152 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Eisenstein Series ◽  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Local Density ◽  
Local Densities ◽  
Quadratic Lattices

Let L, M be regular quadratic lattices over Zp. The local density αp(L, M) is an important invariant in the theory of representation of quadratic forms and they appear naturally in Fourier coefficients of Eisenstein series. In spite of the importance we knew little except the case when either L = M or rk L = 1 and M is unimodular. Evaluating them is a laborious task.

scholarly journals Wavefront dislocations reveal the topology of quasi-1D photonic insulators

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Clément Dutreix ◽  
Matthieu Bellec ◽  
Pierre Delplace ◽  
Fabrice Mortessagne
Keyword(s):  
Local Density ◽  
Topological Defects ◽  
Wave Functions ◽  
Topological Classification ◽  
Local Density Of States ◽  
Topological Phase Transition ◽  
Interference Fringes ◽  
Classical Wave ◽  
Phase Singularities

AbstractPhase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation – a 2D phase singularity – in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

scholarly journals Uniformizing the Moduli Stacks of Global G-Shtukas

2019 ◽  
Author(s):  
E Arasteh Rad ◽  
Urs Hartl
Keyword(s):  
Finite Type ◽  
Moduli Spaces ◽  
Function Field ◽  
Group Scheme ◽  
Base Change ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Moduli Stacks ◽  
Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

scholarly journals MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

2016 ◽  
Vol 230 ◽  
pp. 18-34 ◽  
Author(s):  
MARCO ANTEI ◽  
MICHEL EMSALEM
Keyword(s):  
Finite Number ◽  
Fundamental Group ◽  
Galois Group ◽  
Closed Subset ◽  
Valuation Ring ◽  
Group Scheme ◽  
Discrete Valuation Ring ◽  
Generic Fiber ◽  
Natural Morphism ◽  
Tannakian Category

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

ON FINITE GALOIS STABLE ARITHMETIC GROUPS AND THEIR APPLICATIONS

2008 ◽  
Vol 07 (06) ◽  
pp. 773-783
Author(s):  
EKATERINA S. KHREBTOVA ◽  
DMITRY MALININ
Keyword(s):  
Quantitative Result ◽  
Explicit Construction ◽  
Integral Representations ◽  
Arithmetic Groups ◽  
Group Schemes ◽  
Natural Operation ◽  
Rings Of Integers ◽  
Representations Of Finite Groups ◽  
Finiteness Theorems ◽  
Quadratic Lattices

We prove the existence and finiteness theorems for integral representations stable under Galois operation. An explicit construction of the realization fields for representations of finite groups stable under the natural operation of the Galois group is given. We also compare the representations over fields and the rings of integers, and give a quantitative result on the rarity of integral Galois stable representations. There is a series of related conjectures and applications to arithmetic algebraic geometry, finite flat group schemes, positive definite quadratic lattices and Galois cohomology.

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