Abelian Extensions Research Articles

Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

Suppose E is an elliptic curve over and χ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that . Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell–Weil rank grows in abelian extensions of . Using this heuristic, we find a large class of infinite abelian extensions F where we expect E(F) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.

Local abelian Kato-Parshin reciprocity law: A survey

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

AbstractTwisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

Matrix extension of multidimensional dispersionless integrable hierarchies

We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax-Sato equations, matrix equations on the background of vector fields and the dressing scheme. Reductions, construction of solutions and connections to geometry are discussed. We consider separately a case of Abelian extension, for which the Riemann-Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose formula in the curved space.

Research on Splitting Isomorphism of Leibniz Algebra and Non-Abelian expansion

In this study, Leibniz algebras and the derivations and properties of Leibniz algebras were given, respectively. The stable automorphism group of explicit splitting extension was calculated via the stable automorphism group of Abelian extension of finite group splitting. Based on the stable automorphism group of the splitting extension studied, the non-Abelian extension and the second order non-Abelian co-homology group of Leibniz algebra were investigated in detail according to the stable automorphism group of the splitting extension.

Deformations of thermo-algebras and of Fock spaces on a ring, and open quantum systems

Using a hyperbolic ring based formalism, it is shown that special unitary groups correspond to thermo-algebras for closed quantum systems, while for open systems, the full unitary algebra is the suitable underlying structure. Specifically the groups U(2) and U(1, 1) are realized as the dynamical symmetry groups for open quantum systems through continuous tensorial products; these unitary groups are obtained by mean of abelian extensions through U(1)-factors, from the standard thermo-algebras SU(2), and SU(1, 1).

The commutative inverse semigroup of partial abelian extensions

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of and a normal subgroup H of G, we prove that induces a unital partial action of G/H on the subalgebra of invariants of S such that is an -partial Galois extension of Second, assuming that G is abelian, we construct a commutative inverse semigroup whose elements are equivalence classes of -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between and T(G, A), where ρ is a congruence on and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of reduces to the case where G is cyclic. The set of idempotents of is also investigated.

On 3-Hom-Lie-Rinehart algebras

We introduce the notion of 3-Hom-Lie-Rinehart algebras and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an A-split abelian extension and the equivalence classes of A-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.

On the coefficient-choosing game

Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to $3$ or greater than $8$) with no roots in the maximal abelian extension of $\mathbb{Q}$.

Constructing abelian extensions with prescribed norms

Given a number field K K , a finite abelian group G G and finitely many elements α 1 , … , α t ∈ K \alpha _1,\ldots ,\alpha _t\in K , we construct abelian extensions L / K L/K with Galois group G G that realise all of the elements α 1 , … , α t \alpha _1,\ldots ,\alpha _t as norms of elements in L L . In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.

Genus fields of Kummer ℓn-cyclic extensions

We give a construction of the genus field for Kummer [Formula: see text]-cyclic extensions of rational congruence function fields, where [Formula: see text] is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer [Formula: see text]-cyclic extension. Finally, we study the extension [Formula: see text], for [Formula: see text], [Formula: see text] abelian extensions of [Formula: see text].

Quandle cohomology, extensions and automorphisms

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.

Capitulation abélienne des groupes de classes de rayons

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields.

Lattices from Abelian extensions and error-correcting codes

A construction of laminated lattices of full diversity in odd dimensions d with 3≤d≤15 is presented. The technique, which uses a combination of number fields and error-correcting codes, consists essentially of two steps: In the first, the Abelian number field F of degree d and prime conductor p, where p is a prime congruent to 1 modulo d, is considered. In the second, the lattice is obtained as the canonical embedding (Minkowski homomorphism) of a ℤ-submodule of 𝔒F, the ring of integers of F. The submodule is defined by the parity-check matrices of a Reed–Solomon code over GF(p) and a suitably chosen linear code, typically either binary or over ℤ∕4ℤ, the ring of integers modulo 4.

On the quasitriangular structures of abelian extensions of ℤ2

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of by for an abelian group G. We prove that there are only two forms of them and we get a complete list of all universal -matrices of the generalized Kac-Paljutkin algebra (see Section 2 for the definition).

Complex multiplication and Brauer groups of K3 surfaces

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains Br(X‾)GK for any K3 surface X/K that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.

Coderivations, abelian extensions and cohomology of Lie coalgebras

For any Lie coalgebra with a Lie comodule, a cohomology is constructed by using a subcomplex of the Chevalley–Eilenberg cochain complex of the dual Lie algebra. It is shown that the first (resp. the second) order cohomology group is the space of outer coderivations (resp. the space of equivalent classes of abelian extensions of the Lie coalgebra by its comodule).

Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors

Each \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\) with \(p\)-ring class rank \(\varrho_f\) mod \(f\) is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet \(\mathbf{M}(K_f)=\lbrack(N_{c,i})_{1\le i\le m(c)}\rbrack_{c\mid f}\) of dihedral fields \(N_{c,i}\) with various conductors \(c\mid f\) having \(p\)-multiplicities \(m(c)\) over \(K\) such that \(\sum_{c\mid f}\,m(c)=\frac{p^{\varrho_f}-1}{p-1}\). The advanced viewpoint of classifying the entire collection \(\mathbf{M}(K_f)\), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields. The actual construction of the multiplet \(\mathbf{M}(K_f)\) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.

Exact sequences in the cohomology of a Lie superalgebra extension

Let be an abelian extension of the Lie superalgebra . In this article we consider the problems of extending endomorphisms of and lifting endomorphisms of to certain endomorphisms of . We connect these problems to the cohomology of with coefficients in through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild–Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.