Showing papers on "Ring of integers published in 2005"

Unicity of Types for Supercuspidal Representations of GLN

Abstract: Let $F$ be a non-Archimedean local field, with the ring of integers $\mathfrak{o}_F$. Let $G = \mathrm{GL}_N(F)$, $K = \mathrm{GL}_N (\mathfrak{o}_F)$, and $\pi$ be a supercuspidal representation of $G$. We show that there exists a unique irreducible smooth representation $\tau$ of $K$, such that the restriction to $K$ of a smooth irreducible representation $\pi '$ of $G$ contains $\tau$ if and only if $\pi '$ is isomorphic to $\pi \otimes \chi \circ \det$, where $\chi$ is an unramified quasicharacter of $F^{\times}$. Moreover, we show that $\pi$ contains $\tau$ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence.

Bounded generation of SL(n,A) (after D. Carter, G. Keller and E. Paige)

Abstract: We present unpublished work of DCarter, GKeller, and EPaige on bounded generation in special linear groups Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a localization O_S) If n = 2, assume that A has infinitely many units We show there is a finite-index subgroup H of SL(n,A), such that every matrix in H is a product of a bounded number of elementary matrices We also show that if T is in SL(n,A), and T is not a scalar matrix, then there is a finite-index, normal subgroup N of SL(n,A), such that every element of N is a product of a bounded number of conjugates of T For n > 2, these results remain valid when SL(n,A) is replaced by any of its subgroups of finite index

Division-ample sets and the Diophantine problem for rings of integers

Abstract: R´´ Nous demontrons que le dixieme probleme de Hilbert pour un anneau d'entiers dans un corps de nombres K admet une reponse negative si K satisfaitdeux conditions arithmetiques (existence d'un ensemble dit division-ample et d'une courbe ellip- tique de rang un sur K). Nous lions les ensembles division-ample ` l'arithmetique des varietes abeliennes. Abstract. We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satis- fies two arithmetical conditions (existence of a so-called division- ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.

Characteristic elements in noncommutative iwasawa theory

Abstract: Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their Hasse-Weil L-functions are not only described by the (one-variable) main conjecture corresponding to the cyclotomic Zp-extension kcyc of k, but also by the (two-variable) main conjecture corresponding to the extension k∞ = k(Ep∞) which arises by adjoining the p-power division points Ep∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for non-CM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic p-adic L-function this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic p-adic L-function, viz as an element of the first K-group K1(ΛT ) ∼= ΛT /[Λ × T ,Λ × T ] of a localization ΛT of the usual Iwasawa algebra Λ = Λ(G) of the Galois group G = G(k∞/k). Here, for a ring R, we denote by R× its group of units. By the Weil-pairing, kcyc is contained in k∞ and we put H = G(k∞/kcyc) and Γ = G(kcyc/k). Furthermore we write m(H) for the kernel of the canonical surjective ring homomorphism

Wandering domains and nontrivial reduction in non-Archimedean dynamics

Abstract: Let K be a non-archimedean field with residue field k, and suppose that k is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions φ ∈ K(z) and Rivera-Letelier’s notion of nontrivial reduction. First, if φ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of φ has wandering components by any of the usual definitions of “components of the Fatou set”. Second, we show that if k has characteristic zero and K is discretely valued, then the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate. The theory of complex dynamics in dimension one, founded by Fatou and Julia in the early twentieth century, concerns the action of a rational function φ ∈ C(z) on the Riemann sphere P(C) = C∪{∞}. Any such φ induces a natural partition of the sphere into the closed Julia set Jφ, where small errors become arbitrarily large under iteration, and the open Fatou set Fφ = P (C) \ Jφ. There is also a natural action of φ on the connected components of Fφ, taking a component U to φ(U), which is also a connected component of the Fatou set. In 1985, using quasiconformal methods, Sullivan [32] proved that φ ∈ C(z) has no wandering domains; that is, for each component U of Fφ, there are integers M ≥ 0 and N ≥ 1 such that φ(U) = φ(U). We refer the reader to [1, 13, 24] for background on complex dynamics. Recall that a non-archimedean metric on a space X is a metric d which satisfies the ultrametric triangle inequality d(x, z) ≤ max{d(x, y), d(y, z)} for all x, y, z ∈ X. In the past two decades, beginning with a study of linearization at fixed points by Herman and Yoccoz [19], there have been a number of investigations of non-archimedean dynamics; for a small sampling, see [4, 5, 6, 10, 20, 27, 28, 31]. It is natural to ask which properties of complex dynamics extend to the non-archimedean setting and which do not. We fix the following notation throughout this paper. K a complete non-archimedean field with absolute value | · | K an algebraic closure of K CK the completion of K OK the ring of integers {x ∈ K : |x| ≤ 1} of K k the residue field of K OCK the ring of integers {x ∈ CK : |x| ≤ 1} of CK k the residue field of CK Date: July 30, 2003. 2000 Mathematics Subject Classification. Primary: 11S80; Secondary: 37F10, 54H20.

Optimal norm form integer space-time codes for two antenna MIMO systems

Abstract: In this paper, we introduce the definitions of a full diversity integer generating matrix and the corresponding norm form space-time code for MIMO systems. Subject to a power constraint, we characterize all full diversity integer generating matrices with the first three largest gains in the Gaussian integer ring and the Eisenstein integer ring for two transmitter antennas. Using this generating matrix family to separately design space-time codes layer by layer for two transmitter antenna and two receiver antenna MIMO systems, we obtain the optimal norm form integer space-time codes both in the Gaussian integer ring and the Eisenstein integer ring in the sense of maximizing the minimum determinant of codeword matrices. As a consequence, we prove that the golden code constructed by Dayal and Varanasi, Belfiore, Rekaya and Viterbo is optimal in the Gaussian integer ring. Also, we find the optimal code in the Eisentein integer ring, the coding gain of which is greater than that of the golden code.

Realizable Galois module classes for tetrahedral extensions

Abstract: Let k be a number field with ring of integers $\mathfrak{O}_k$ , and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N / k with group isomorphic to $\Gamma$ , the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ . We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$ . This refines a result of Godin and Sodaigui ( J. Number Theory 98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$ . To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k .

Multiple planar coincidences with N-fold symmetry

Abstract: Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.

Euler characteristics of arithmetic groups

Abstract: We develop a general method for computing the homological Euler characteristic of finite index subgroups G of GL_m(O_K) where O_K is the ring of integers in a number field K. With this method we find, that for large, explicitly computed dimensions m, the homological Euler characteristic of finite index subgroups of GL_m(O_K) vanishes. For other cases, some of them very important for spaces of multiple polylogarithms, we compute non-zero homological Euler characteristic. With the same method we find all the torsion elements in GL_3(Z) up to conjugation. Finally, our method allows us to obtain a formula for the Dedekind zeta function at -1 in terms of the ideal class set and the multiplicative group of quadratic extensions of the base ring.

Power Integral Bases in the Family of Simplest Quartic Fields

Abstract: Several authors have considered the infinite parametric family of simplest quartic fields Kt = ℚ(ξ). In this paper, we explicitly give all generators of power integral bases in the ring of integers ℤ K of Kt assuming that t 2+16 is not divisible by an odd square. We use a well known general algorithm for calculating power integral bases in quartic fields.

Zeta functions over $\mathbf{F}_1$

Abstract: We show basic properties of zeta functions over the one element field starting from an algebraic set over the integer ring. We calculate several examples and we investigate special values via the associated $K$-group identified as the stable homotopy group of spheres.

Traces of singular moduli on Hilbert modular surfaces

Abstract: Suppose that p ≡ 1 (mod 4) is a prime, and that OK is the ring of integers of K := Q( √ p). A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface (h × h)/SL2(OK) are weight 2 holomorphic modular forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singular moduli over these intersection points are often weakly holomorphic weight 2 modular forms. For the singular moduli of J1(z) = j(z) − 744, we explicitly determine these generating functions using classical Weber functions, and we factorize their “norms” as products of Hilbert class polynomials. We also explicitly compute all such generating functions in the “SL2(Z) case” for the primes p = 5, 13, and 17.

Dynamical systems in unramified or totally ramified extensions of a \mathfrak{p}-adic field

Abstract: We use local field theory to study a special class of discrete dynamical systems, where the function being iterated is a polynomial whose coefficients belong to the ring of integers in a -adic field.

Counting Strings with Given Elementary Symmetric Function Evaluations II: Circular Strings

Abstract: Let $\alpha$ be a string over an alphabet that is a finite ring, $R$. The $k$th elementary symmetric function evaluated at $\alpha$ is denoted $T_k(\alpha)$. In a companion paper we studied the properties of $\mathbf{S}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, the set of length $n$ strings for which $T_i(\alpha) = \tau_i$. Here we consider the set, $\mathbf{L}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, of equivalence classes under rotation of aperiodic strings in $\mathbf{S}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, sometimes called Lyndon words. General formulae are established and then refined for the cases where $R$ is the ring of integers $\Z{q}$ or the finite field $\F{q}$.

Integer and Rational Solutions to Polynomial Equations

Abstract: A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational solutions to $y^2=P(x)$ is facilitated by another elliptic curve with integral coefficients. The problem of counting is described by two elliptic curves and a map between them.

A spherical CR structure on the complement of the figure eight knot with discrete holonomy

Abstract: We describe a general geometrical construction of spherical CR structures. We construct then spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in $PU(2,1,{\bf Z}[\omega])$ and $PU(2,1,{\bf Z}[i])$ respectively. These are the same ring of integers appearing in the real hyperbolic geometry of the corresponding links.

Some examples of Hecke algebras over 2-dimensional local fields

Abstract: Let K be a local non-archimedian field, F=K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical (and Iwahori) Hecke algebras for representations of the group G(F) and its central extension by means of K*. For instance our spherical Hecke algebra corresponds to the subgroup G(A) where A is the subring $\calO_K((t))$ where $\calO_K\subset K$ is the ring of integers. It turns out that for generic level the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

On invariants related to non-unique factorizations in block monoids and rings of algebraic integers

Abstract: Let K be a number field, R its ring of integers and H the set of non-zero principal ideals of R. For each positive integer k the set Bk(H) C H denotes the set of principal ideals for which the associated block has at most k different factorizations. For the counting functions associated to these sets asymp­ totic formulae are known. These formulae involve constants that just depend on the ideal class group G of R. Starting from a known combinatorial description for these constants, we use tools from additive group theory, in particular the notion of Davenport's constant and a classical addition theorem, to investigate them. We determine their precise value in case G is an elementary group or a cyclic group of prime power order. For arbitrary G we derive (explicit) lower bounds .

The Iwasawa Invariants and the Higher K-Groups Associated to Real Quadratic Fields

Abstract: Using fast algorithms, we compute the Iwasawa invariants of Q( √ f, ζp) in the range 1

Power integral bases for real cyclotomic fields

Abstract: We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

Higher class groups of generalized eichler orders

Abstract: In this paper we study the possible torsion in even-dimensional higher class groups Cl 2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl 2n (Λ) can only occur for primes p dividing prime ideals ℘ of 𝒪 F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.

Almost-periodic solutions for a second order abstract cauchy problem

Abstract: Let M be a number field of degree m with ring of integers \bZ_M . Let F\in\bZ_M[X,Y] be a form of degree n such that F(X,1) has distinct roots. Let\break G\in\bZ[X,Y] be an arbitrary polynomial of degree k . Assuming that k\le n-2m\pl 1 if all roots of F^{(i)}(X,1) (1\le i\le n) are complex and k\le n-4m\pl 1 otherwise, we provide an efficient algorithm for finding all solutions X,Y\in\bZ_M , \max\b(\overline{|X|},\overline{|Y|}\,\b)\ki C of the inequality \overline{\b|F(X,Y)\b|\!}\,\le c \cdot \overline{\b|G(X,Y)\b|\!}\,. We provide numerical examples with m=3 and C=10^{100} .

The k th-order quasi-generalized bent functions over ring Z p

Abstract: In this paper, we propose a new class of logical functions over residue ring of integers modulo p, where p is a prime. The magnitudes of the Chrestenson Spectra for this kind of functions, called as kth-order quasi-generalized Bent functions, take only two values—0 and a nonzero constant. By using the relationships between Chrestenson spectra and the autocorrelation functions for logical functions over ring Zp, we present some equivalent definitions of this kind of functions. In the end, we investigate the constructions of the kth-order quasi-generalized Bent functions, including the typical method and the recursive method from the technique of number theory.

Semiconjugacy of quasiperiodic flows and finite index subgroups of multiplier groups

Abstract: It will be shown that ifis a quasiperiodic flow on the n-torus that is algebraic, if is a flow on the n-torus that is smoothly conjugate to a flow generated by a constant vector field, and ifis smoothly semiconjugate to , then is a quasiperiodic flow that is algebraic, and the multiplier group of is a finite index subgroup of the multiplier group of �. This will partially establish a conjecture that asserts that a quasiperiodic flow on the n-torus is algebraic if and only if its multiplier group is a finite index subgroup of the group of units of the ring of integers in a real algebraic number field of degree n.

On Locally Finite Modules

Abstract: This paper is devoted to the study of locally finite modules M, i.e., modules whose finitely generated submodules are finite (as sets). In particular, we study rings which have faithful locally finite modules, for example, the polynomial rings ℤ[x] and F[x], where F is a finite field. Our results generalize the properties of Abelian torsion groups and the ring of integers.

Quaternary Constructions for the Binary Single- Error-Correcting Codes of Julin, Best and Others

Abstract: Certain nonlinear binary single-error-correcting codes found by Julin, Best and others have simple descriptions as codes over the ring of integers modulo 4