scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
Author(s):  
Bruce W. Jordan ◽  
Allan G. Keeton ◽  
Bjorn Poonen ◽  
Eric M. Rains ◽  
Nicholas Shepherd-Barron ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Opposite Direction ◽  
Abelian Varieties ◽  
Sufficient Conditions ◽  
Higher Dimensional ◽  
Free Left ◽  
Necessary And Sufficient ◽  
Finitely Presented ◽  
Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

scholarly journals Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function inLp(·)Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Muhammad Sarwar ◽  
Ghulam Murtaza ◽  
Irshaad Ahmed
Keyword(s):  
Integral Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Fractional Integral Operator ◽  
Lebesgue Spaces ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Decreasing Functions ◽  
Hardy Operators

One-weight inequalities with general weights for Riemann-Liouville transform andn-dimensional fractional integral operator in variable exponent Lebesgue spaces defined onRnare investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions inLp(x)spaces.

scholarly journals Complete addition laws on abelian varieties

2012 ◽  
Vol 15 ◽  
pp. 308-316 ◽  
Author(s):  
Christophe Arene ◽  
David Kohel ◽  
Christophe Ritzenthaler
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Galois Group ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Rational Points ◽  
Absolute Galois Group ◽  
Projective Embedding ◽  
Complete Set

AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

scholarly journals Perfect hypercomplex algebras: Semi-tensor product approach

2021 ◽  
Vol 1 (4) ◽  
pp. 177-187
Author(s):  
Daizhan Cheng ◽  
◽  
Zhengping Ji ◽  
Jun-e Feng ◽  
Shihua Fu ◽  
...  
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Characteristic Functions ◽  
Hypercomplex Numbers ◽  
3 Dimensional ◽  
Zero Sets ◽  
Higher Dimensional ◽  
Product Approach ◽  
Necessary And Sufficient

<abstract><p>The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.</p></abstract>

scholarly journals Controllability of networked higher-dimensional systems with one-dimensional communication

2017 ◽  
Vol 375 (2088) ◽  
pp. 20160215 ◽  
Author(s):  
Lin Wang ◽  
Xiaofan Wang ◽  
Guanrong Chen
Keyword(s):  
Network Topology ◽  
Linear Time ◽  
Sufficient Conditions ◽  
External Control ◽  
One Dimensional ◽  
Networked System ◽  
Linear Time Invariant ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Time Invariant

In this paper, the state controllability of networked higher-dimensional linear time-invariant dynamical systems is considered, where communications are performed through one-dimensional connections. The influences on the controllability of such a networked system are investigated, which come from a combination of network topology, node-system dynamics, external control inputs and inner interactions. Particularly, necessary and sufficient conditions are presented for the controllability of the network with a general topology, as well as for some special settings such as cycles and chains, which show that the observability of the node system is necessary in general and the controllability of the node system is necessary for chains but not necessary for cycles. Moreover, two examples are constructed to illustrate that uncontrollable node systems can be assembled to a controllable networked system, while controllable node systems may lead to uncontrollable systems even for the cycle topology. This article is part of the themed issue ‘Horizons of cybernetical physics’.

Topological Properties of a Class of Higher-dimensional Self-affine Tiles

2019 ◽  
Vol 62 (4) ◽  
pp. 727-740
Author(s):  
Guotai Deng ◽  
Chuntai Liu ◽  
Sze-Man Ngai
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Higher Dimensional ◽  
Necessary And Sufficient

AbstractWe construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

scholarly journals Computing isogenies between abelian varieties

2012 ◽  
Vol 148 (5) ◽  
pp. 1483-1515 ◽  
Author(s):  
David Lubicz ◽  
Damien Robert
Keyword(s):  
Abelian Variety ◽  
Time Complexity ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Null Point ◽  
Constant Number ◽  
Weil Pairing ◽  
Base Field ◽  
Rational Expression ◽  
Higher Dimensional

AbstractWe describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ)g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

scholarly journals The Manin constant in the semistable case

2018 ◽  
Vol 154 (9) ◽  
pp. 1889-1920 ◽  
Author(s):  
Kęstutis Česnavičius
Keyword(s):  
Vector Space ◽  
Elliptic Curve ◽  
Abelian Varieties ◽  
Higher Dimensional ◽  
Special Cases ◽  
Modular Parametrization ◽  
De Rham ◽  
Neron Models ◽  
General Relations ◽  
Néron Models

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

Relatively free dimonoids and bar-units

2021 ◽  
pp. 1-13
Author(s):  
Anatolii V. Zhuchok
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Case ◽  
Free Left ◽  
Necessary And Sufficient

This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.

CONSTRUCTION AND PROPERTIES OF QUASI RICCI FLAT SPACES

1986 ◽  
Vol 01 (04) ◽  
pp. 997-1007 ◽  
Author(s):  
GUY BONNEAU ◽  
FRANÇOIS DELDUC
Keyword(s):  
String Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Real ◽  
Ricci Flat ◽  
Compact Case ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Lagrangian Densities ◽  
Torsion Free

We look for the necessary and sufficient conditions for a generalized torsion-free nonlinear σ-model to be one-loop finite. The corresponding metrics are not only Ricci flat ones, but also a larger class we call “quasi Ricci flat” spaces. We give expressions for the corresponding Lagrangian densities in the real and Kähler cases. In the latter, the manifold is shown to be proper, complete and nonhomogeneous. Unfortunately, in the compact case, relevant for string theory, these quasi Ricci flat manifolds become Ricci flat ones.

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