Quotient Modulo Research Articles

Linear Complexity of r-Ary Sequences Derived from Euler Quotient Modulo pq

Linear Complexity of r-Ary Sequences Derived from Euler Quotient Modulo pq

L-functions of elliptic curves modulo integers

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see [9]). If E/K is an elliptic curve with nonconstant j-invariant defined over a function field K of characteristic p≥5, we know that its L-function L(T,E/K) is a polynomial in Z[T] (see [5, p. 11]). Inspired by Schoof, we study the reduction of L(T,E/K) modulo integers. We obtain three main results. Firstly, if E/K has non-trivial K-rational N-torsion for some positive integer N coprime with p, we extend a formula for L(T,E/K)modN due to Hall (see [4, p. 133, Theorem 4]) to all quadratic twists Ef/K with f∈K×∖K×2. Secondly, without any condition on the 2-torsion subgroup of E(K), we give a formula for the quotient modulo 2 of L-functions of any two quadratic twists of E/K. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and, under some assumption, find in most cases the exact analytic rank of these twists. We also illustrate that in favourable situations, our second main result allows one to compute much more efficiently L(T,Ef/K)mod2 than an algorithm of Baig and Hall (see [1]). Finally, we use our formulas to compute some degree 2 L-functions directly.

Normal edge-transitive Cayley graphs and Frattini-like subgroups

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist of those automorphisms of G which leave C invariant. We define an Aut(G;C)-invariant normal subgroup Φ(G;C) of G which has the property that, for any Aut(G;C)-invariant normal set of generators for G, if we remove from it all the elements of Φ(G;C), then the remaining set is still an Aut(G;C)-invariant normal generating set for G. The subgroup Φ(G;C) contains the Frattini subgroup Φ(G) but the inclusion may be proper. The Cayley graph Cay(G,C) is normal edge-transitive if Aut(G;C) acts transitively on the pairs {c,c−1} from C. We show that, for a normal edge-transitive Cayley graph Cay(G,C), its quotient modulo Φ(G;C) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ(G;C) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi.

Linear Complexity of Binary Threshold Sequences Derived from Generalized Polynomial Quotient with Prime-Power Modulus

In this paper, firstly we extend the polynomial quotient modulo an odd prime [Formula: see text] to its general case with modulo [Formula: see text] and [Formula: see text]. From the new quotient proposed, we define a class of [Formula: see text]-periodic binary threshold sequences. Then combining the Legendre symbol and Euler quotient modulo [Formula: see text] together, with the condition of [Formula: see text], we present exact values of the linear complexity for [Formula: see text], and all the possible values of the linear complexity for [Formula: see text]. The linear complexity is very close to the period and is of desired value for cryptographic purpose. Our results extend the linear complexity results of the corresponding [Formula: see text]-periodic binary sequences in earlier work.

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

GRADIENT RICCI SOLITONS WITH SEMI-SYMMETRY

We prove that a semi-symmetric 3-dimensional gradient Ricci soliton is locally isometric to a space form <TEX>$\mathbb{S}^3$</TEX>, <TEX>$\mathbb{H}^3$</TEX>, <TEX>$\mathbb{R}^3$</TEX> (Gaussian soliton); or a product space <TEX>$\mathbb{R}{\times}\mathbb{S}^2$</TEX>, <TEX>$\mathbb{R}{\times}\mathbb{H}^2$</TEX>, where the potential function depends only on the nullity.

Relations between tautological cycles on Jacobians

We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Further we lift a result of Herbaut and van der Geer–Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk’s operator \mathcal{D} lifts to the tautological subalgebra of CH(J).

Spectral synthesis in the Fourier algebra and the Varopoulos algebra

The objects of study in this paper are sets of spectral synthesis for the Fourier algebra $A(G)$ of a locally compact group and the Varopoulos algebra $V(G)$ of a compact group with respect to submodules of the dual space. Such sets of synthesis are characterized in terms of certain closed ideals. For a closed set in a closed subgroup $H$ of $G,$ the relations between these ideals in the Fourier algebras of $G$ and $H$ are obtained. The injection theorem for such sets of synthesis is then a consequence. For the Fourier algebra of the quotient modulo a compact subgroup, an inverse projection theorem is proved. For a compact group, a correspondence between submodules of the dual spaces of $A(G)$ and $V(G)$ is set up and this leads to a relation between the corresponding sets of synthesis.

Heller triangulated categories

Let E be a Frobenius category. Let E denote its stable category. The shift functor on E induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in E. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying still a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation onE, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.

The stable derived category of a noetherian scheme

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

Equivariant classification of Gorenstein open log del Pezzo surfaces with finite group actions

We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups.

On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed as a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M. We will study this conjecture for modules M e R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanley's Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

Varieties of uniserial representations IV. Kinship to geometric quotients

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\Bbb S}$ a finite sequence of simple left $\Lambda$-modules. In [6, 9], quasiprojective algebraic varieties with accessible affine open covers were introduced, for use in classifying the uniserial representations of $\Lambda$ having sequence ${\Bbb S}$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations' -- in a sense to be made precise -- to geometric quotients of the classical varieties $\operatorname{Mod-Uni}({\Bbb S})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. To some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered uniserial varieties can be embedded. These tools are then applied towards the third objective, concerning the existence of geometric quotients: We prove that $\operatorname{Mod-Uni}({\Bbb S})$ has a geometric quotient by the $GL$-action precisely when the uniserial variety has a geometric quotient modulo a certain natural algebraic group action, in which case the two quotients coincide. Our main results are exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.

Characterizing a class of simply presented modules by relation arrays

Simply presented modules and Warfield modules are described in a class of mixed modules \(\cal H \) with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary rank. Analogous to a result of Warfield it is shown that the mixed modules of torsion-free rank one are in some sense the building blocks of such modules. The results extend our previous work describing this class by relation arrays which are a natural outgrowth of their basic generating systems. Moreover, an intimate connection is shown between relation arrays and the indicators of modules. Furthermore, we prove two realizability results one of which is analogous to a theorem of Megibben for mixed modules of torsion-free rank one. One gives necessary and sufficient conditions on when a relation array of a module can realize an indicator of finite-type while the other shows that an admissable indicator can be realized by a simply presented module of torsion-free rank one.

Nonreduced extended affine root systems of nullity 3

In 1985 K. Saito [S] introduced the concept of an extended affine root system (EARS). His study of these root systems was motivated by his interest in singularities. Later in [AABGP], this notion played an important role in the study of extended affine Lie algebras. Saito classified the EARS's of nullity ≤ 2 which have the further property that the quotient modulo a "marking" is reduced. In [AABGP], a construction was given of all EARS's, and this was used to give a classification of EARS's of reduced type. However, when the EARS was not reduced, the isomorphism problem for the construction is quite difficult and the classification was only done for EARS'S withnullity ≤ 2. The present paper extends this classification to nullity 3.

Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals

The method of dehomogenizing graded rings has been used successfully in algebraic geometry, e.g., a determinental ring is a dehomogenization of a Schubert cycle. We extend this method to noncommutative graded rings, dehomogenizing suitably graded rings to Zariski filtered rings and deriving, in a very elementary way, homological properties related to Auslander regularity and the Gorenstein property for noncommutative rings. As an application we study the lifting of such properties from a quotient modulo an invertible ideal.

ON MINIMAL MODELS OF ALGEBRAIC CURVES

Let be an odd prime number. Consider the algebraic curves (normalizations of their projective closures): Let be a primitive th root of 1. The Galois group acts on the minimal models of these curves over . This idea is used here to study their minimal models over . The action of , passage to the quotient modulo this action, the resolution of singularities on the quotients, and the contraction of exceptional curves of genus 1 are described. All of this leads to minimal models of the indicated curves over .Bibliography: 6 titles.

Indecomposable representations of the Lorentz algebra in an angular momentum basis

Indecomposable (i.e. reducible but not completely reducible) representations of the (complex) algebra so(3,1) of the Lorentz group are analysed. A master representation of so(3,1) is obtained on the space of its universal enveloping algebra Omega . The basis chosen for so(3,1) is the 'angular momentum' basis, and for Omega a 'natural basis' with respect to the angular momentum basis of so(3,1). The master representation induces representations on a space Omega - which is obtained from Omega as quotient space modulo certain ideals. These representations have the property that rho (h3) is diagonal. A change of basis is made from the natural basis to the angular momentum basis and the representations on Omega - are analysed in this new basis. The indecomposable (as well as the irreducible) representations which are obtained have the property that their so(3) content consists of infinite-dimensional indecomposable and irreducible so(3) representations. Certain of the indecomposable so(3,1) representations have additional invariant subspaces which lead to quotient spaces which are of finite dimension. These quotient spaces then carry the familiar finite-dimensional representations of the Lorentz algebra. It is shown that the standard theory of representations of the Lorentz algebra is contained in this analysis as a special case of representations which are induced on certain quotient spaces of the space Omega +.

Complexity of ?wild? matrix problems and of isomorphism of algebras and graphs

It is shown that isomorphism of semisimple algebras over an algebraically closed field is recognized in polynomial time. The polynomial equivalence of isomorphism of graphs and isomorphism of algebras (over an algebraically closed field) with zero square of the radical and commutative quotient modulo the radical is proved. A series of problems about the complexity of matrix problems and isomorphism of algebras are posed.