Isomorphism Classes Research Articles

Entanglement of elliptic curves upon base extension

Fix distinct primes [Formula: see text] and [Formula: see text] and let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. The [Formula: see text]-entanglement type of [Formula: see text] over [Formula: see text] is the isomorphism class of the group [Formula: see text]. The size of this group measures the extent to which the image of the mod [Formula: see text] Galois representation attached to [Formula: see text] fails to be a direct product of the mod [Formula: see text] and mod [Formula: see text] images. In this paper, we study how the [Formula: see text]-entanglement group varies over different base fields. We prove that for each prime [Formula: see text] dividing the greatest common divisor of the size of the mod [Formula: see text] and [Formula: see text] images, there are infinitely many fields [Formula: see text] such that the entanglement over [Formula: see text] is cyclic of order [Formula: see text]. We also classify all possible [Formula: see text]-entanglement types that can occur as the base field [Formula: see text] varies.

Cycle Switching in Steiner Triple Systems of Order 19

ABSTRACTCycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order (an ), yielding another . This relationship may be represented by an undirected graph. An admits cycles of lengths and . In the particular case of , it is known that the full switching graph, allowing the switching of cycles of any length, is connected. We show that if we restrict switching to only one of the possible cycle lengths, in all cases, the switching graph is disconnected (even if we ignore those s, which have no cycle of the given length). Moreover, in a number of cases we find intriguing connected components in the switching graphs, which exhibit unexpected symmetries. Our method utilizes an algorithm for determining connected components in a very large implicitly defined graph which is more efficient than previous approaches, avoiding the necessity of computing canonical labelings for a large proportion of the systems.

The Counting of Isomorphism Classes of Mixed Graphs

The Counting of Isomorphism Classes of Mixed Graphs

5-Dimensional Malcev-Like Algebras

The five-dimensional anti-commutative algebras having an analogous family of flags of subalgebras as the solvable Malcev algebras form the class of Malcev-like algebras. Recently a classification of the binary Lie algebras in this class is achieved. Our investigation extends this result. We determine Malcev-like algebras over a field K of characteristic zero. These algebras are extensions of K by a 4-dimensional nilpotent Lie algebra and simultaneously semidirect sums of the two-dimensional non-abelian Lie algebra and an abelian algebra. We find normal forms of their multiplications and describe their isomorphism classes.

NON-LOG LIFTABLE LOG DEL PEZZO SURFACES OF RANK ONE IN CHARACTERISTIC FIVE

Abstract Building upon the classification by Lacini, we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that such a surface is always constructed from the Du Val del Pezzo surface of Dynkin type $2[2^4]$ . Furthermore, We show that the Kawamata–Viehweg vanishing theorem for ample $\mathbb {Z}$ -Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.

A classification of nilpotent compatible Lie algebras

Working over an arbitrary field of characteristic different from 2, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension 4. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension 3, and 12 isomorphism classes, 6 one-parameter families and one 2-parameter family in dimension 4.

Hom-associative magmas with applications to Hom-associative magma algebras

Let $X$ be a magma, that is a set equipped with a binary operation, and consider a function $\alpha : X \to X$. We say that $X$ is Hom-associative if, for all $x,y,z \in X$, the equality $\alpha(x)(yz) = (xy) \alpha(z)$ holds. For every isomorphism class of magmas of order two, we determine all functions $\alpha$ making $X$ Hom-associative. Furthermore, we find all such $\alpha$ that are endomorphisms of $X$. We also consider versions of these results where the binary operation on $X$ and the function $\alpha$ only are partially defined. We use our findings to construct numerous examples of two-dimensional Hom-associative as well as multiplicative magma algebras.

A filtered generalization of the Chekanov–Eliashberg algebra

We define a new algebra associated to a Legendrian submanifold \Lambda of a contact manifold of the form \mathbb{R}_{t}\times \mathrm{W} , called the planar diagram algebra , and denote it by \mathrm{PDA}(\Lambda, \mathcal{P}) . It is a non-commutative, filtered, differential graded algebra whose filtered stable tame isomorphism class is an invariant of \Lambda together with a partition \mathcal{P} of its connected components. When \Lambda is connected, \mathrm{PDA} is the Chekanov–Eliashberg algebra. In general, the \mathrm{PDA} differential counts holomorphic disks with multiple positive punctures using a combinatorial framework inspired by string topology.

Average Analytic Ranks of Elliptic Curves over Number Fields

AbstractWe give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$ , when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

Distinguishing local isomorphism classes in quasicrystals by high-order harmonic spectroscopy

Electron diffraction spectroscopy is a fundamental tool for investigating quasicrystal structures, which unveils the quasiperiodic long-range order. Nevertheless, it falls short in effectively distinguishing separate local isomorphism classes. This is a long outstanding problem. Here, we study the high-order harmonic generation in two-dimensional generalized Penrose quasicrystals to optically resolve different local isomorphism classes. The results reveal that: (i) harmonic spectra from different parts of a quasicrystal are identical, even though their atomic arrangements vary significantly. (ii) The harmonic yields of diverse local isomorphism classes exhibit variations, providing a way to distinguish local isomorphism classes. (iii) The rotational symmetry of harmonic yield can serve as a characteristic of quasicrystal harmonics and is consistent with the orientation order. Our results not only pave the way for confirming the experimental reproducibility of quasicrystal harmonics and identifying quasicrystal local isomorphism classes, but also shed light on comprehending electron dynamics influenced by the vertex environments.

A family of non-isomorphic maximal function fields

The problem of understanding whether two given function fields are isomorphic is well-known to be difficult, particularly when the aim is to prove that an isomorphism does not exist. In this paper we investigate a family of maximal function fields that arise as Galois subfields of the Hermitian function field. We compute the automorphism group, the Weierstrass semigroup at some special rational places and the isomorphism classes of such function fields. In this way, we show that often these function fields provide in fact examples of maximal function fields with the same genus, the same automorphism group, but that are not isomorphic.

Houghton-like groups from “shift-similar” groups

We introduce and study shift-similar groups G \le \operatorname{Sym}(\mathbb{N}) , which play an analogous role in the world of Houghton groups that self-similar groups play in the world of Thompson groups. We also introduce Houghton-like groups H_{n}(G) arising from shift-similar groups G , which are an analog of Röver–Nekrashevych groups from the world of Thompson groups. We prove a variety of results about shift-similar groups and these Houghton-like groups, including results about finite generation and amenability. One prominent result is that every finitely generated group embeds as a subgroup of a finitely generated shift-similar group, in contrast to self-similar groups, where this is not the case. This establishes in particular that there exist uncountably many isomorphism classes of finitely generated shift-similar groups, again in contrast to the self-similar situation.

Recovering the Picard group of quadratic algebras from Wood’s binary quadratic forms

Let [Formula: see text] be a scheme such that [Formula: see text] is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over [Formula: see text], how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over [Formula: see text] and isomorphism classes of pairs [Formula: see text] where [Formula: see text] is a quadratic algebra over [Formula: see text] and [Formula: see text] is an invertible [Formula: see text]-module. Unexpectedly, examples suggest that a refinement of Wood’s bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over [Formula: see text]; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.

Group gradings on triangularizable algebras

Group gradings on triangularizable algebras

The pointed p-groups on a block algebra

The pointed p-groups on a block algebra

Generic properties of topological groups

Abstract We study generic properties of topological groups in the sense of Baire category. First, we investigate countably infinite groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. Recently, I. Goldbring, S. Kunnawalkam Elayavalli, and Y. Lodha proved that every isomorphism class is meager among countably infinite groups. In contrast, it follows from the work of W. Hodges on model-theoretic forcing that there exists a comeager isomorphism class among countably infinite abelian groups. We present a new elementary proof of this result. Then, we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.

Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

The aim here is to describe all isomorphism classes of SU(n+1)-equivariant Hermitian holomorphic vector bundles on the complex projective space CPn. If G⊂SU(n+1) is the isotropy subgroup of a chosen point x0∈CPn, and ρ:G⟶GL(V) is a unitary representation, we obtain SU(n+1)-equivariant holomorphic Hermitian vector bundles on CPn. Next, given any v∈End(Vρ)⊗(Tz00,1CPn)∗ satisfying certain conditions, a new structure of an SU(n+1)-equivariant holomorphic Hermitian vector bundle on this underlying C∞ holomorphic Hermitian bundle is obtained. It is shown that all SU(n+1)-equivariant holomorphic Hermitian vector bundles on CPn arise this way.

Principal bundle structure of the space of metric measure spaces

We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group ${\mathbb{R}}_+$ of positive real numbers on $\mathcal{X}$ , which has a one-point metric measure space, say $*$ , as only one fixed-point. We prove that the ${\mathbb{R}}_+$ -action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of non-trivial and locally trivial principal ${\mathbb{R}}_+$ -bundle over the quotient space. Our bundle ${\mathbb{R}}_+ \to \mathcal{X}_* \to \mathcal{X}_*/{\mathbb{R}}_+$ is a curious example of a non-trivial principal fibre bundle with contractible fibre. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$ , where we completely determine the structure of the fixed-point set of the ${\mathbb{R}}_+$ -action on the compactification.

On the homotopy groups of the automorphism groups of Cuntz–Krieger algebras

In this paper, we first present the homotopy groups of the automorphism groups of Cuntz–Krieger algebras in terms of the underlying matrices defining the Cuntz–Krieger algebras. We also show that the homotopy groups are complete invariants of the isomorphism classes of Cuntz–Krieger algebras. As a result, the isomorphism class of a Cuntz–Krieger algebra is completely determined by the group structure of its weak extension group and strong extension group.

Antiassociative magmas

AbstractA magma S is said to be antiassociative if $$(ab)c \ne a(bc)$$ ( a b ) c ≠ a ( b c ) holds for any elements a, b, c of S. Antiassociative magmas lie at the opposite pole from associative ones, known as semigroups. In this paper, we study antiassociative magmas focusing on their properties and examples that can be seen from Cayley tables. We provide a test for the antiassociativity of a finite magma and give some general methods for constructing antiassociative magmas. We also characterize, describe, and count all antiassociative magma structures on a 3-element set, all their isomorphism classes, and all classes of equivalent magmas of this type.