Showing papers on "Ring of integers published in 2011"

The catenary degree of Krull monoids I

Abstract: Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a 2 H and each two factorizations z, z 0 of a, there exist factorizations z = z0, . . . , zk = z 0 of a such that, for each i 2 (1, k), zi arises from zi−1 by replacing at most N atoms from zi−1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) � 4.

The geometry of the Gauss–Picard modular group

Abstract: We give a construction of a fundamental domain for $${{\rm PU}(2,1,\mathbb{Z} [i])}$$ , that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers $${\mathbb{Z} [i]}$$ . We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

The basic geometry of Witt vectors. II: Spaces

Abstract: This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, “p-typical” Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the “big” Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of Buium’s formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.

Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Abstract: Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

Perfect cuboids and irreducible polynomials

Abstract: The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$, and $u$ would mean the non-existence of perfect cuboids. This irreducibility is conjectured and then verified numerically for approximately 10000 instances of $a$, $b$, and $u$.

Irreducibility criteria of Schur-type and Pólya-type

Abstract: Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Polya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Polya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

Automorphic products on unitary groups

Abstract: The dissertation provides a construction for Borcherds products on unitary groups of signature (1,q). The starting point for this is the multiplicative lifting due to R. E. Borcherds. He employs the singular theta-correspondence to construct a lifting, which takes as inputs weakly holomorphic vector valued modular forms, transforming under the Weil-representation of SL(2,Z) for a quadratic lattice, and lifts these to meromorphic automorphic forms for an arithmetic subgroup of O(2,n). The resulting functions have expansions as infinite products and take their zeros and poles along Heegner divisors. In order to transfer this result to unitary groups, we construct an embedding between the symmetric domain of the unitary group and that of an orthogonal group, respectively. This embedding is compatible with the complex structures of either symmetric domain and a suitable choice of cusps. The main result is the construction of Borcherds products, on unitary groups of signature (1,q). In this setting we prove a result which is analogous to that of Borcherds. As in the case of orthogonal groups, the infinite products thus constructed have their zeros and poles on Heegner divisors. Here, the role of the quadratic lattice is taken by a hermitian lattice, which we assume to have as multiplier system the ring of integers of an imaginary quadratic number field. Further, we study the behavior of these automorphic products on the boundary of the symmetric domain. It turns out that the values taken on the boundary points can be interpreted as CM-values of generalized eta-products. In the finial chapter, we construct examples for the unitary group SU(1,1) and unimodular lattices, which in this case are simply hyperbolic planes over the rings of integers of imaginary quadratic number fields. In this case, the resulting products can be viewed as meromorphic elliptic modular forms on the (classical) complex upper half-plane.

Pillars and towers of quadratic transformations

Abstract: Infinite pillars of quadratic transformations are used to describe residue fields of subrings of finitely generated ring extensions of the ring of integers. Towers whose underlying quadratic transformations are finite pillars or nonpillars are employed for the construction of basic dicritical divisors.

New Fully Homomorphic Encryption over the Integers.

Abstract: We first present a fully homomorphic encryption scheme over the integers, which modifies the fully homomorphic encryption scheme in [vDGHV10]. The security of our scheme is merely based on the hardness of finding an approximate-GCD problem over the integers, which is given a list of integers perturbed by the small error noises, removing the assumption of the sparse subset sum problem in the origin scheme [vDGHV10]. Then, we construct a new fully homomorphic encryption scheme, which extends the above scheme from approximate GCD over the ring of integers to approximate principal ideal lattice over the polynomial integer ring. The security of our scheme depends on the hardness of the decisional approximate principle ideal lattice polynomial (APIP), given a list of approximate multiples of a principal ideal lattice. At the same time, we also provide APIP-based fully homomorphic encryption by introducing the sparse subset sum problem. Finally, we design a new fully homomorphic encryption scheme, whose security is based on the hardness assumption of approximate lattice problem and the decisional SSSP.

A generalization of Eisenstein–Schönemann irreducibility criterion

Abstract: One of the results generalizing Eisenstein Irreducibility Criterion states that if $${\phi(x) = a_nx^n\,{+} \,a_{n-1}x^{n-1} \,{+} \,\cdots\,{+} \,a_0}$$ is a polynomial with coefficients from the ring of integers such that a s is not divisible by a prime p for some $${s \, \leqslant \, n}$$ , each a i is divisible by p for $${0 \, \leqslant \, i \, \leqslant \, s-1}$$ and a 0 is not divisible by p 2, then $${\phi(x)}$$ has an irreducible factor of degree at least s over the field of rational numbers. We have observed that if $${\phi(x)}$$ is as above, then it has an irreducible factor g(x) of degree s over the ring of p-adic integers such that g(x) is an Eisenstein polynomial with respect to p. In this paper, we prove an analogue of the above result for a wider class of polynomials which will extend the classical Schonemann Irreducibility Criterion as well as Generalized Schonemann Irreducibility Criterion and yields irreducibility criteria by Akira et al. (J Number Theory 25:107–111, 1987).

The homological torsion of SL_2 of the imaginary quadratic integers

Abstract: Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

Indecomposable 1-morphisms of \dot{U}^+_3 and the canonical basis of U_q^+(sl_3)

Abstract: We compute the indecomposable objects of \dot{U}^+_3 - the category that categorifies the positive half of the quantum sl_3, and we decompose an arbitrary object into indecomposable ones. On decategorified level we obtain the Lusztig's canonical basis of the positive half U^+_q(sl_3) of the quantum sl_3. We also categorify the higher quantum Serre relations in U_q^+(sl_3), by defining a certain complex in the homotopy category of $\dot{U}^+_3$ that is homotopic to zero. We work with the category $\dot{U}^+_3$ that is defined over the ring of integers. This paper is based on the (extended) diagrammatic calculus introduced to categorify quantum groups.

Accessing the cohomology of discrete groups above their virtual cohomological dimension

Abstract: We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kramer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

A quantitative aspect of non-unique factorizations: the narkiewicz constants

Abstract: Let K be an algebraic number field with non-trivial class group G and let be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let Fk (x) denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk (x) behaves, for x → ∞, asymptotically like x(log x)-1+1/|G|(log log x)Nk(G). We study Nk (G) with new methods from Combinatorial Number Theory.

Lambda actions of rings of integers

Abstract: Let O be the ring of integers of a number field K. For an O-algebra R which is torsion free as an O-module we define what we mean by a Lambda_O-ring structure on R. We can determine whether a finite etale K-algebra E with Lambda_O-ring structure has an integral model in terms of a Deligne-Ribet monoid of K. This a commutative monoid whose invertible elements form a ray class group.

An alternative description of the Drinfeld p-adic half-plane

Abstract: We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a non-archimedean local field F represents a moduli problem of polarized O_F-modules with an action of the ring of integers O_E in a quadratic extension E of F. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL_2(F) and SU(C)(F) for a two-dimensional split hermitian space C for E/F.

Small solutions to systems of polynomial equations with integer coe-cients

Abstract: The paper discusses a series of conjectures due to A. Tyszka aiming to describe boxes in which there exists at least one solution to a system of polynomial equations with integer coe‐cients. A proof of the bound valid in the linear case is given. 1 Two basic questions When facing systems of equations whose solutions are hard to determine, one is satisfled to determine (or at least estimate) the number and the size of solutions. A satisfactory answer could be an algorithm, if a deflnite formula is unavailable. These questions are completely answered only for univariate polynomials over the ring of integers or the fleld of rational, real or complex numbers. Many important results, such as Falting’s result on rational points on irreducible algebraic curves of genus at least 2, ensures the flniteness of the solution set to speciflc systems without giving any hint on its cardinality. A great deal of mathematics appeared as a result of attempts to solve such ‡ i : ¶ ;

Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

Abstract: Given the ring of integers OK of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, OK) such that OKG, the OK-span of G, coincides with M(n, OK), the ring of (n × n)-matrices over OK? The answer is known if n is an odd prime In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, OK) such that OKG = M(2m, OK)

Multiply monogenic orders

Abstract: Let O be an order in an algebraic number field K, i.e., a ring with quotient field K which is contained in the ring of integers of K. The order O is called monogenic, if it is of the shape Z[w], i.e., generated over the rational integers by one element. By a result of Gy\H{o}ry (1976), the set of w with Z[w]=O is a union of finitely many equivalence classes, where two elements v,w of O are called equivalent if v+w or v-w is a rational integer. An order O is called k times monogenic if there are at least k different equivalence classes of w with Z[w]=O, and precisely k times monogenic if there are precisely k such equivalence classes. It is known that every quadratic order is precisely one time monogenic, while in number fields of degree larger than 2, there may be non-monogenic orders. In this paper we study orders which are more than one time monogenic. Our first main result is, that in any number field K of degree at least 3 there are only finitely many three times monogenic orders. Next, we define two special types of two times monogenic orders, and show that there are number fields K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K, we prove that K has only finitely many two times monogenic orders which are not of these types. We give some immediate applications to canonical number systems. Further, we prove extensions of our results for domains which are monogenic over a given domain A of characteristic 0 which is finitely generated over Z.

Quantitative spectral gap for thin groups of hyperbolic isometries

Abstract: Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

Localisation and colocalisation of KK-theory

Abstract: The localisation of an R-linear triangulated category $\mathcal{T}$ at S −1 R for a multiplicatively closed subset S is again triangulated, and related to the original category by a long exact sequence involving a version of $\mathcal{T}$ with coefficients in S −1 R/R. We examine these theories and, under some assumptions, write the latter as an inductive limit of $\mathcal{T}$ with torsion coefficients. Our main application is the case where $\mathcal{T}$ is equivariant bivariant K-theory and R the ring of integers.

On the maximality of secret data ratio in CPTE schemes

Abstract: Based on the ring of integers modulo 2r, Chen-Pan-Tseng (2000) introduced a block-based scheme (CPT scheme) which permits in each block F of size m×n of a given binary image B to embed r =⌊log2(k+1)⌋ secret bits by changing at most two entries of F, where k=mn . As shown, the highest number of embedded secret bits for at most two bits to be changed in each block of k positions of F in any CPT-based schemes is rmax=⌊log2(1+k (k+1)/2)⌋, approximately 2r-1, twice as much as r asymptotically, and this can reached approximately in our CPTE1 scheme by using modules on the ring Z2 of integers modulo 2. A new modified scheme-CPTE2 to control the quality of the embedded blocks, in the same way as Tseng-Pan's method (2001), is established. Approximately, CPTE2 scheme gives 2r-2 embedded bits in F, twice as much as r-1 bits given by Tseng-Pan' scheme, while the quality is the same.

Enumerating maximal definite quadratic forms of bounded class number over Z in n >= 3 variables

Abstract: In this paper we give an algorithm for enumerating all primitive (positive) definite maximal Z-valued quadratic forms Q in n >= 3 variables with bounded class number h(Q) = 3 variables. Using this we determine that there are exactly 115 primitive positive definite maximal Z-valued quadratic forms in n >= 3 variables of class number one, and produce a list of them. In a future paper we will complete this chain of ideas by extending these algorithms to allow the enumeration of all primitive maximal totally definite O_F-valued quadratic lattices of rank n >= 3, where O_F is the ring of integers of any totally real number field F.

Non-unique factorization and principalization in number fields

Abstract: We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible factorizations of an element in the ring of integers of a number field, and give a combinatorial description for the number of such factorizations. In certain cases, we show how quadratic forms can explicitly provide all such factorizations.

Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

Abstract: We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.

Algebraic integers as special values of modular units

Abstract: Let ϕ(τ )= η( 1 (τ + 1)) 2 / √ 2π exp{ 1 πi}η(τ + 1), where η(τ ) is the Dedekind eta function. We show that if τ0 is an imaginary quadratic argument and m is an odd integer, then √ mϕ(mτ0)/ϕ(τ0) is an algebraic integer dividing √ m. This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and θK is an element of K with Im(θK ) > 0 which generates the ring of integers of K over Z, we find a sufficient condition on m which ensures that √ mϕ(mθK )/ϕ(θK ) is a unit.

Nonunique factorization and principalization in number fields

Abstract: Following what is basically Kummer’s relatively neglected approach to nonunique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms can be used to explicitly produce all irreducible factorizations of n.