Non-maximal Orders Research Articles

Natural homomorphism of Witt rings of a certain cubic order

In this paper, we provide an example of a nonmaximal order in a cubic number field [Formula: see text] whose Witt ring is embedded into the Witt ring of [Formula: see text].

Toeplitz algebras of semigroups

To each submonoid P P of a group we associate a universal Toeplitz C ∗ \mathrm {C}^* -algebra T u ( P ) \mathcal {T}_u(P) defined via generators and relations; T u ( P ) \mathcal {T}_u(P) is a quotient of Li’s semigroup C ∗ \mathrm {C}^* -algebra C s ∗ ( P ) \mathrm {C}^*_s(P) and they are isomorphic iff P P satisfies independence. We give a partial crossed product realization of T u ( P ) \mathcal {T}_u(P) and show that several results known for C s ∗ ( P ) \mathrm {C}^*_s(P) when P P satisfies independence are also valid for T u ( P ) \mathcal {T}_u(P) when independence fails. At the level of the reduced semigroup C ∗ \mathrm {C}^* -algebra T λ ( P ) \mathcal {T}_\lambda (P) , we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of P P , generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of T λ ( P ) \mathcal {T}_\lambda (P) is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over P P with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on P P for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the a x + b ax+b -monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails.

Torsion points and Galois representations on CM elliptic curves

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

Computing the ideal class monoid of an order

There are well known algorithms to compute the class group of the maximal order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order $R$ in a finite product of number fields $K$. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain $\mathbb{Q}$-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.

Rationality of Darmon points over genus fields of non-maximal orders

Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor $c\geq 1$ twisted by quadratic characters of $G_c^+=Gal(H_c^+/F)$, where $H_c^+$ is the strict ring class field of $F$ of conductor $c$, come from rational points on the elliptic curve defined over $H_c^+$.

Explicit computations in Iwasawa theory

We give two algorithms to compute layers of the anticyclotomic ${\bf Z}_3$-extension of an imaginary quadratic field. The first is based on complex multiplication techniques for nonmaximal orders; the second is based on Kummer theory. As an illustration of our results, we use the mirroring principle to derive results on the structure of class groups of nonmaximal orders.

Witt functor of a quadratic order

Abstract The main theorem of the paper gives an example of a non-maximal order 𝓞 in a quadratic number field K such that the homomorphism W𝓞 → WK of Witt rings is injective.

Representation fields for orders of small rank

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.

Examples of Ore extensions which are maximal orders whose based rings are not maximal orders

Let R be a prime Goldie ring and (σ,δ) be a skew derivation on R. It is well known that if R is a maximal order, then the Ore extension R[x;σ,δ] is a maximal order. It was a long standing open question that the converse is true or not in case σ≠1 and δ≠0.We give an example of non-maximal order R with a skew derivation (σ,δ) on R (σ≠1,δ≠0) such that R[x;σ,δ] is a maximal order.

위계집단에서 효율적인 암호계의 안전성에 관한 소고

A cryptography for enforcing hierarchic groups in a system where hierarchy is represented by a partially ordered set was introduced by Akl et al. But the key generation algorithm of Akl et al. is infeasible when there is a large number of users. To overcome this shortage, in 1985, MacKinnon et al. proposed a paper containing a condition which prevents cooperative attacks and optimizes the assignment. In 2005, Kim et al. proposed the key management systems for using one-way hash function, RSA algorithm, poset dimension and Clifford semigroup in the context of modern cryptography, the key management system using Clifford semigroup of imaginary quadratic non-maximal orders. We, in this paper, show that Kim et al. cryptosystem is insecure in some reasons and propose a revised cryptosystem.

Representation fields for commutative orders

Un corps de représentation pour un ordre non maximal ℌ dans une algèbre centrale simple est un sous-corps du corps de classes spinoriel d’ordres maximaux qui détermine l’ensemble de genres spinoriels d’ordres maximaux qui contiennent un conjugué de ℌ. Un ordre non maximal ne possède pas forcément un corps de représentation. Dans ce travail, nous montrons que chaque ordre commutatif a un corps de représentation F et nous donnons une formule pour F. Le résultat principal est prouvé pour des algèbres simples centrales sur des corps globaux arbitraires.

On the Security of Cryptosystems Based on Imaginary Quadratic Class Semigroups

In this paper, we propose a new discrete logarithm problem(DLP) based on the class semigroups of imaginary quadratic non-maximal orders using the special character of non-invertible ideal and analysis its security. To do this, we first explain the mathematical background explicitly and prove some properties of Cls (O) which relate to constructing the DLP and guaranteeing the security. To test the security of the proposed DLP, we compare the class number of the maximal order with that of the non-maximal order and investigate the unique factorization problems of ideals between class groups of the maximal orders and class semigroups of non-maximal orders to ensure the security of the cryptosystem.

Relative spinor class fields: A counterexample

It is known that classes of indefinite quadratic forms in a genus are classified by the Galois group of a spinor class field [4]. Hsia has proved the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. Spinor class fields can also be used to classify conjugacy classes of maximal orders in a central simple algebra. In [1] we left open the issue of whether for every fixed given non-maximal order $$\mathfrak{H}$$ in a central simple division algebra there exists a representation field L with the property that $$\mathfrak{H}$$ embeds into a given maximal order if and only if the corresponding element of the Galois group is trivial on L. In this work we give a negative answer to this question for central simple division algebras of dimension ≥ 32. The case of non-division algebras is also treated by replacing the phrase embeds into by is contained in a conjugate of. As a byproduct of the techniques used in this paper we compute the representation field of an Eichler order in a quaternion algebra.

ON THE PUBLIC KEY CRYPTOSYSTEMS OVER CLASS SEMIGROUPS OF IMAGINARY QUADRATIC NON-MAXIMAL ORDERS

In this paper we will propose the methods for finding the non-invertible ideals corresponding to non-primitive quadratic forms and clarify the structures of class SEMIGROUPS of imaginary quadratic orders which were given by Zanardo and Zannier [8], and we will give a general algorithm for calculating power of ideals/classes via the Dirichlet composition of quadratic forms which is applicable to cryptography in the class semigroup of imaginary quadratic non-maximal order and revisit the cryptosystem of Kim and Moon [5] using a Zanardo and Zannier [8]'s quantity as their secret key, in order to analyze Jacobson [7]'s revised cryptosystem based on the class semigroup which is an alternative of Kim and Moon [5]'s.

Decomposition of primes in non-maximal orders

Let R be a Dedekind ring with quotient field K, and let F (x) ∈ R[x] be an irreducible primitive polynomial. Let α be a root of F and L = K(α). If F is monic, then R[α] is a monogenic R-order of L. If F is not monic, then one can construct an R-order RF of L that suitably generalizes R[α]. We show that, even if F is not monic, the classical invariants of RF can be derived directly from the polynomial F , precisely as in the case of monogenic orders.

Cale Bases in Algebraic Orders

Let R be a non-maximal order in a finite algebraic number field with integral closure R ¯. Although R is not a unique factorization domain, we obtain a positive integer N and a family 𝒬 (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of 𝒬 for each x∈R coprime with the conductor of R. Moreover, this property holds for each nonzero x∈R when the natural map Spec(R ¯)→Spec(R) is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.

Towards Practical Non-interactive Public Key Cryptosystems Using Non-maximal Imaginary Quadratic Orders

We present a new non-interactive public-key distribution system based on the class group of a non-maximal imaginary quadratic order Cl( Δp). The main advantage of our system over earlier proposals based on (Z}/nZ)* [25,27] is that embedding id information into group elements in a cyclic subgroup of the class group is easy (straight-forward embedding into prime ideals suffices) and secure, since the entire class group is cyclic with very high probability. Computational results demonstrate that a key generation center (KGC) with modest computational resources can set up a key distribution system using reasonably secure public system parameters. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of ΔpeΔ1 p2. We present an algorithm for computing discrete logarithms in Cl(Δp) by reducing the problem to computing discrete logarithms in Cl(Δ1) and either F*p or F*p2. Our algorithm is a specific case of the more general algorithm used in the setting of ray class groups [5]. We prove—for arbitrary non-maximal orders—that this reduction to discrete logarithms in the maximal order and a small number of finite fields has polynomial complexity if the factorization of the conductor is known.

Tamagawa numbers for motives with (non-commutative) coefficients

Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple \mathbb Q -algebra A . We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the A -equivariant L -function of M . This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order \mathfrak A in A for which there exists a 'projective \mathfrak A -structure' on M . The existence of such a structure is guaranteed if \mathfrak A is a maximal order, and also occurs in many natural examples where \mathfrak A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.

Cyclic subgroups of ideal class groups in real quadratic orders

The primary purpose of this paper is to provide general sucient conditions for any real quadratic order to have a cyclic subgroup of order n 2 N in its ideal class group. This generalizes results in the literature, including some seminal classical works. This is done with a simpler approach via the interplay between the maximal order and the non-maximal orders, using the underlying infrastructure via the continued fraction algorithm. Numerous examples and a concluding criterion for non-trivial class numbers are also provided. The latter links class number one criteria with new prime-producing quadratic polynomials. 1. Notation and preliminaries. We will be considering arbitrary real quadratic orders, so we ®rst introduce the notions of arbitrary discriminants and radicands. Let D0 6ˆ 1 be a square-free integer, and set 0 ˆ D0 if D0 1 mod 4†, 4D0 otherwise. Then 0 is called a fundamental discriminant with associated fundamental radicand D0. Let f 2 N, and set ˆ f 2 0. Then ˆ D if D0 1 mod 4† and f is odd, 4D otherwise, is a discriminant with conductor f , and associated radicand D ˆ f =2† D0 if D0 1 mod 4† and f is even, f 2 D0 otherwise, having underlying fundamental discriminant 0 with associated fundamental radicand D0. Let be a discriminant with associated radicand D. Then ! ˆ 1‡  D p †=2 if ˆ D 1 mod 4†,  D p if 0 mod 4†, is called the principal surd associated with . This will provide the canonical basis element for our orders. First we need notation for a Z-module: ‰ ; Š ˆ x‡ y : x; y 2 Z ; Glasgow Math. J. 41 (1999) 197±206. # Glasgow Mathematical Journal Trust 1999. Printed in the United Kingdom where , 2 K ˆ Q  p † ˆ Q  D0 p †, the real quadratic ®eld of discriminant 0 and radicand D0. For this reason, fundamental discriminants are often called ®eld discriminants. In particular, if we set O ˆ ‰1; ! Š, then this is an order in K. Also, the index jO 0 : O j ˆ f is the conductor associated with , where O 0 is the maximal order in K, sometimes called the ring of integers of K. In other words, the maximal order in K is the order with conductor f ˆ 1, having square-free radicand D0 and fundamental discriminant 0. We also need to be able to distinguish those Z-modules that are ideals in O ; (see [1, pp. 9±30]). Theorem 1.1 (Primitive ideals and norms). Let be a discriminant, and let I 6ˆ 0† be a Z-submodule of O . Then I has a representation of the form I ˆ ‰a; b‡ c! Š, where a; c 2 N and b 2 Z with 0 b 0 is a discriminant and I is an O -ideal with N I † 0 be a discriminant with associated radicand D ˆ t ‡ r for t 2 N and jrj ˆ 1; 4. If I 1 in C , with N I † <  p =2, then one of the following holds. 1. N I † ˆ t=2, where r ˆ 1 and t is even. 2. N I † ˆ 4, where r ˆ 4 and t is even. 3. N I † ˆ ty 2, where r ˆ y4 and t is odd. 4. N I † ˆ 1. A formula for the class number of an order is given by h ˆ h 0 0 f †=u; 1:1† where f is the conductor associated with ,

Criteria for unique factorization in integral domains

Let R be an integral domain. In this paper, we introduce a sequence of factorization properties which are weaker than the classical UFD criteria. We give several examples of atomic nonfactorial monoids which satisfy these conditions, but show for several classes of integral domains of arithmetical interest that these factorization properties force unique factorization. In particular, we show that if R satisfies any of our properties and is a Krull domain with finite divisor class group, a nonmaximal order in an algebraic number field, or a generalized Cohen-Kaplansky domain, then R in fact must be factorial.