Commutative Schemes Research Articles

Every $BT_1$ group scheme appears in a Jacobian

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over $k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

Pseudo-rigid p-torsion finite flat commutative group schemes

Let p be a prime number and k a perfect field of characteristic p. In the present paper, we study deformations of finite flat commutative group schemes over k to the ring W of Witt vectors with coefficients in k. We prove that, for a given principally quasi-polarizable p-torsion finite flat commutative group scheme over k, it holds that the group scheme is pseudo-rigid — i.e., roughly speaking, has a unique, up to isomorphism over W, deformation to W — if and only if the group scheme is superspecial.

On association schemes with multiplicities 1 or 2

Inspired by the work of Amitsur [1] on finite groups whose irreducible characters all have degree (multiplicity) 1 or 2, in this paper we study association schemes whose irreducible characters all have multiplicity 1 or 2. We will first show that the general case can be reduced to commutative association schemes. Then for commutative association schemes with multiplicities 1 or 2, we prove that their Krein parameters are all rational integers. Using automorphism groups of association schemes, we obtain a characterization and classification of those commutative association schemes all valencies and multiplicities of which are 1 or 2 in terms of Cayley schemes.

Full Level Structure On Some Group Schemes

We give a definition of full level structure on group schemes of the form \(G\times G\), where G is a finite flat commutative group scheme of rank p over a \({\mathbb {Z}}_p\)-scheme S or, more generally, a truncated p-divisible group of height 1. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.

Motivic Hypercohomology Solutions in Field Theory and Applications in H-States

Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\textup{DM}_{\textup {gm}}(k)$,  which is of the motivic objects whose image is under $\textup {Spec}(k)$  that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\mathbb{Q}(q)$,  are obtained when cohomology groups of the isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$,   can be vanished for  $p>\textup{dim}(Y)$.  We observe also the Beilinson-Soul$\acute{e}$ vanishing  conjectures where we have the vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \ \textup{if} \ \ p\leq0,$ and $q>0$,   which confirms the before established. Then survives a hypercohomology $\mathbb{H}^{q}(X,\mathbb{Q})$. Then its objects are in $\textup{Spec(Sm}_{k})$.  Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\Sigma$.  Likewise, will be proved that $\mathrm{H}^{\vee}$,  has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$,  on the general linear group with $k=\mathbb{C}$.  A physics re-interpretation of the superposing, to the dual of the spectrum $\mathrm{H}^{\vee}$,  whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ is the corresponding deformed derived category for densities $\mathrm{h} \in \mathrm{H}$, in quantum field theory.

Use of SHDM in commutative watermarking encryption

SHDM stands for Sphere-Hardening Dither Modulation and is a watermarking algorithm based on quantizing the norm of a vector extracted from the cover work. We show how SHDM can be integrated into a fully commutative watermarking-encryption scheme and investigate implementations in the spatial, DCT, and DWT domain with respect to their fidelity, robustness, capacity, and security of encryption. The watermarking scheme, when applied in the DCT or DWT domain, proves to be very robust against JPEG/JPEG2000 compression. On the other hand, the spatial domain-based approach offers a large capacity. The increased robustness of the watermarking schemes, however, comes at the cost of rather weak encryption primitives, making the proposed CWE scheme suited for low to medium security applications with high robustness requirements.

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.

Disjoint weighing matrices

The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search leads to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes.

Logarithmic Structures of Fontaine-Illusie. II ---Logarithmic Flat Topology

This is Part II of the author's paper Logarithmic structures of Fontaine-Illusie. We discuss log flat topology and log flat descent. We study the first log flat cohomology $H^1(X_{\log, \text{fl}}, G)$ for various sheaves of groups $G$, for example, $G=GL_n$, finite flat commutative group schemes, the log multiplicative group $M^{\text{gp}}$, etc.

On structures of quasi-thin table algebras and applications to association schemes

In this paper we develop a structure theory for a class of table algebras, i.e. the so-called quasi-thin table algebras. They are direct generalizations of quasi-thin association schemes. Although this class of table algebras is rather close to finite groups, its structural properties are quite different from those of finite groups. The thin radical and thin residue give rise to two group structures for every table algebra. They play the central role in our discussions. Applications to association schemes are also discussed, and some new results about quasi-thin association schemes are obtained. In particular, we will give necessary and sufficient conditions under which a commutative quasi-thin association scheme has multiplicities 1 or 2. We will need these results in the study of the characterization and classification of double quasi-thin association schemes in [22].

The higher-dimensional Contou-Carrère symbol and commutative group schemes

The higher-dimensional Contou-Carrère symbol and commutative group schemes

Quantum protocol for privacy preserving Hamming distance problem of DNA sequences

The privacy preserving Hamming distance problem of DNA sequences is defined and, by using a quantum commutative encryption scheme, a novel quantum protocol for solving it is proposed. A 0-1 coding scheme of DNA sequence is constructed, which transforms the privacy preserving Hamming distance problem of DNA sequences into privacy preserving Hamming distance problem of bit strings. Compared with other related protocols, the efficiency of the proposed protocol is improved, because of entanglement states, joint measurement, unitary operations and even Bell-state measurement are not needed, while only rotation operations and single-state measurement are required, which are easier to be realized with current technology. Furthermore, the correctness and security of the proposed protocol are also analyzed.

Duality for complexes of tori over a global field of positive characteristic

If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.

The higher-dimensional Contou-Carrère symbol and commutative group schemes

The higher-dimensional Contou-Carrère symbol and commutative group schemes

Homogeneous vector bundles over abelian varieties via representation theory

Let A A be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over A A form an abelian category HVec A \textrm {HVec}_A ; the Fourier-Mukai transform yields an equivalence of HVec A \textrm {HVec}_A with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of HVec A \textrm {HVec}_A with the category of finite-dimensional representations of a commutative affine group scheme (the “affine fundamental group” of A A ). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.

Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes

For prime powers q and q+ε where ε∈{1,2}, an affine resolvable design from Fq and Latin squares from Fq+ε yield a set of symmetric designs if ε=2 and a set of symmetric group divisible designs if ε=1. We show that these designs derive commutative association schemes, and determine their eigenmatrices.

On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes

We will give an equation on Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes.

Efficient solution to the millionaires' problem based on asymmetric commutative encryption scheme

AbstractSecure multiparty computation is an important scheme in cryptography and can be applied in various real‐life problems. The first secure multiparty computation problem is the millionaires' problem, and its protocol is an important building block. Because of the less efficiency of public key encryption scheme, most existing solutions based on public key cryptography to this problem are inefficient. Thus, a solution based on the symmetric encryption scheme has been proposed. In this paper, we formally analyse the vulnerability of this solution, and propose a new scheme based on the decisional Diffie‐Hellman assumption. Our solution also uses 0‐encoding and 1‐encoding generated by our modified encoding method to reduce the computation cost. We implement the solution based on symmetric encryption scheme and our protocol. Extensive experiments are conducted to evaluate the efficiency of our solution, and the experimental results show that our solution can be much more efficient and be approximately 8000 times faster than the solution based on symmetric encryption scheme for a 32‐bit input and short‐term security. Moreover, our solution is also more efficient than the state‐of‐the‐art solution without precomputation and can also compare well with the state‐of‐the‐art protocol while the bit length of private inputs is large enough.

Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois coverings

Let f:S′→S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r≥1, let ResG(r):Hr(Se´t,G)→Hr(Se´t′,G) and CoresG(r):Hr(Se´t′,G)→Hr(Se´t,G) be, respectively, the restriction and corestriction maps in étale cohomology induced by f. For certain pairs (f,G), we construct maps αr:KerCoresG(r)→CokerResG(r) and βr:CokerResG(r)→KerCoresG(r) such that αr∘βr=βr∘αr=n. In the simplest nontrivial case, namely when f is a quadratic Galois covering, we identify the kernel and cokernel of βr with the kernel and cokernel of another map CokerCoresG(r−1)→KerResG(r+1). We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme S to that of a quadratic Galois cover S′ of S.