Showing papers on "Ring of integers published in 2013"

The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$

Abstract: Let K be a finite extension over Q2 and OK the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over OK .

Ergodic properties of $k$-free integers in number fields

Abstract: Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers $\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences, equipped with an $\mathscr O_K$-invariant probability measure associated to $\mathscr F_k$. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the work of Cellarosi and Sinai [ J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.

Integer valued polynomials on lower triangular integer matrices

Abstract: Let \(D\) be an integral domain with quotient field \(K\). In this paper we study the algebra of polynomials in \(K[x]\) which map the set of lower triangular \(n\times n\) matrices with coefficients in \(D\) into itself and show that it coincides with the algebra of polynomials whose divided differences of order \(k\) map \(D^{k+1}\) into \(D\) for every \(k< n\). Using this result we describe the polynomial closure of this set of matrices when \(D\) is the ring of integers in a global field.

Higher torsion in the Abelianization of the full Bianchi groups

Abstract: Denote by , with a square-free positive integer, an imaginary quadratic number field, and by its ring of integers. The Bianchi groups are the groups . In the literature, so far there have been no examples of -torsion in the integral homology of the full Bianchi groups, for a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance at the discriminant . Supplementary materials are available with this article.

A New Class of Balanced Near-Perfect Nonlinear Mappings and Its Application to Sequence Design

Abstract: A mapping from ZN to ZM can be directly applied for the design of a sequence of period N with alphabet size M, where ZN denotes the ring of integers modulo N. The nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence. When M is a divisor of N, the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation, but it is not balanced. In this paper, we study balanced near-perfect nonlinear (NPN) mappings applicable for the design of sequence sets with low correlation. We first construct a new class of balanced NPN mappings from Z(p2-p) to Zp for an odd prime p. We then present a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping. By applying it to the new class, we obtain a new optimal FHS set of period p2-p with respect to the Peng-Fan bound, whose FHSs are balanced and optimal with respect to the Lempel-Greenberger bound. Moreover, we construct a low-correlation sequence set with size p, period p2-p, and maximum correlation magnitude p from the new class of balanced NPN mappings, which is asymptotically optimal with respect to the Welch bound.

Tameness Results for Expansions of the Real Field by Groups

Abstract: Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we introduce some basic logical concepts and theorems of o-minimality. In Chapter 2, we prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals. We give a few applications of this result. The main theorem of Chapter 3 is a structure theorem for expansions of the real field by families of restricted complex power functions. We apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. Chapter 4 deals with polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional submanifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d.

Selmer groups as flat cohomology groups

Abstract: Given a prime number $p$, Bloch and Kato showed how the $p^\infty$-Selmer group of an abelian variety $A$ over a number field $K$ is determined by the $p$-adic Tate module. In general, the $p^m$-Selmer group $\mathrm{Sel}_{p^m} A$ need not be determined by the mod $p^m$ Galois representation $A[p^m]$; we show, however, that this is the case if $p$ is large enough. More precisely, we exhibit a finite explicit set of rational primes $\Sigma$ depending on $K$ and $A$, such that $\mathrm{Sel}_{p^m} A$ is determined by $A[p^m]$ for all $p ot \in \Sigma$. In the course of the argument we describe the flat cohomology group $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ of the ring of integers of $K$ with coefficients in the $p^m$-torsion $\mathcal{A}[p^m]$ of the Neron model of $A$ by local conditions for $p ot\in \Sigma$, compare them with the local conditions defining $\mathrm{Sel}_{p^m} A$, and prove that $\mathcal{A}[p^m]$ itself is determined by $A[p^m]$ for such $p$. Our method sharpens the known relationship between $\mathrm{Sel}_{p^m} A$ and $H^1_{\mathrm{fppf}}(O_K, \mathcal{A}[p^m])$ and continues to work for other isogenies $\phi$ between abelian varieties over global fields provided that $\mathrm{deg} \phi$ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve $11A1$ over certain families of number fields.

Wiretap encoding of lattices from number fields using codes over F p

Abstract: We consider the problem of communication over a block fading wiretap channel. It is known that coding for such a channel can be done using nested lattice codes constructed over totally real number fields. In this paper, we propose a method for encoding an integral lattice over the ring of integers of a totally real number field, and study in particular the case of Q(ζp+ζp-1) using a linear code over Fp. This generalizes the well-known Construction A and provides an efficient coset encoding for algebraic lattices.

Quotients of orders in cyclic algebras and space-time codes

Abstract: Let $F$ be a number field with ring of integers $\boldsymbol{O}_F$ and $D$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\boldsymbol{O}_F$-order $\Lambda$ in $D$ by two-sided ideals. We reduce the problem to studying the ideal structure of $\Lambda/q^s\Lambda$, where $q$ is a prime ideal in $\boldsymbol{O}_F$, $s\geq 1$. We study the case where $q$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.

A NOTE ON THE q-ANALOGUE OF KIM'S p-ADIC log GAMMA TYPE FUNCTIONS ASSOCIATED WITH q-EXTENSION OF GENOCCHI AND EULER NUMBERS WITH WEIGHT

Abstract: In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)

On the pullback of an arithmetic theta function

Abstract: In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series $${\widehat{\phi}_{\mathcal C}(\tau)}$$ defined using arithmetic 0-cycles on the moduli space $${\mathcal C}$$ of elliptic curves with CM by the ring of integers $${O_{\kappa}}$$ of an imaginary quadratic field. The second such series $${\widehat{\phi}_{\mathcal M}(\tau)}$$ has weight 3/2 and takes values in the arithmetic Chow group $${\widehat{{\rm CH}}^1(\mathcal{M})}$$ of the arithmetic surface associated to an indefinite quaternion algebra $${B/\mathbb{Q}}$$ . For an embedding $${O_\kappa \rightarrow O_B}$$ , a maximal order in B, and a two sided O B -ideal Λ, there is a morphism $${j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}$$ and a pullback $${j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}$$ . Our main result is an expression for the pullback $${j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}$$ as a linear combination of products of $${\widehat{\phi}_{\mathcal C}(\tau)}$$ ’s and classical weight $${\frac{1}{2}}$$ theta series.

The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers

Abstract: The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

An effective proof of the hyperelliptic Shafarevich conjecture

Abstract: Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.

Asymptotically Optimal Optical Orthogonal Codes With New Parameters

Abstract: Optical orthogonal codes (OOCs) are widely used as spreading codes in optical fiber networks. An (N, w, λa, λc)-OOC with size L is a family of L {0,1}-sequences with length N, weight w, maximum autocorrelation λa, and maximum cross correlation λc. In this paper, we present two new constructions for OOCs with λa=λc=1 which are asymptotically optimal with respect to the Johnson bound. We first construct an asymptotically optimal (Mpn, M, 1,1)-OOC with size (pn-1)/M by using the structure of Zpn, the ring of integers modulo pn, where p is an odd prime with M|p-1, and n is a positive integer. We then present another asymptotically optimal (Mp1...pk, M, 1,1)-OOC with size (p1...pk-1)/M from a product of k finite fields, where pi is an odd prime and M is a positive integer such that M| pi-1 for 1 ≤ i ≤ k. In particular, it is optimal in the case that k=1 and (M-1)2 > p1-1.

Optimal Fraction-Free Routh Tests for Complex and Real Integer Polynomials

Abstract: The Routh test is the simplest and most efficient algorithm to determine whether all the zeros of a polynomial have negative real parts. However, the test involves divisions that may decrease its numerical accuracy and are a drawback in its use for various generalized applications. The paper presents fraction-free forms for this classical test that enhance it with the property that the testing of a polynomial with Gaussian or real integer coefficients can be completed over the respective ring of integers. Two types of algorithms are considered one, named the G-sequence, which is most efficient (as an integer algorithm) for Gaussian integers, and another, named the R-sequence, which is most efficient for real integers. The G-sequence can be used also for the real case, but the R-sequence is by far more efficient for real integer polynomials. The count of zeros with positive real parts for normal polynomials is also presented for each algorithm.

Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

Abstract: Though one can consider Motivic Integration to have quite satisfactory foundations in residue characteristic zero after [11], [12] and [21], much remains to be done in positive residue characteristic. The aim of the present paper is to explain how one can extend the formalism and results from [11] to mixed characteristic. Other aims are to give an axiomatic approach instead of using only the Denef-Pas language, and to extend the formalism of [11] to one with richer angular component maps. Let us start with some motivation. Let K be a fixed finite field extension of Qp with residue field Fq and let Kd denote its unique unramified extension of degree d, for d ≥ 1. Denote by Od the ring of integers of Kd and fix a polynomial H ∈ O1[x1, · · · , xn]. For each d one can consider the Igusa local zeta function

Diophantine quadruples in the ring of integers of the pure cubic field $Q(\sqrt[3]{2})$

Abstract: We show that in the ring of integers of the pure cubic field Q( 3 √ 2) there exits a D(w)-quadruple if and only if w can be represented as a difference of two squares of integers in Q( 3 √ 2).

Normal zeta functions of the Heisenberg groups over number rings I - the unramified case

Abstract: Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups $H(\mathcal{O}_K)$ that are indexed by rational primes which are unramified in $K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.

A universal first-order formula defining the ring of integers in a number field

Abstract: We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ..., x_N) is not 0}. We will use global class field theory and generalize the ideas originating from Koenigsmann's recent result giving a universal first order formula for Z in Q.

On the Probability of Relative Primality in the Gaussian Integers

Abstract: This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{\zeta_{\Q(i)}(2)}$$ where $z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $\zeta_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theorem to arbitrary number fields that are principal ideal domains.

Duality of Preenvelopes and Pure Injective Modules

Abstract: Let $R$ be an arbitrary ring and $(-)^+=\Hom_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.

The Frobenius Problem for Modified Arithmetic Progressions

Abstract: For a set of positive and relatively prime integers A, let ( A) denote the set of integers obtained by taking all nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to ( A). For the modified arithmetic progression A = {a,ha + d,ha + 2d,...,ha + kd}, gcd(a,d) = 1, we determine the largest integer g(A) that does not belong to ( A), and the number of integers n(A) that do not belong to ( A). We also determine the set of integers S ⋆ (A) that do not belong to ( A) which, when added to any positive integer in ( A), result in an integer in ( A). Our results generalize the corresponding results for arithmetic progressions.

Distance between conjugate algebraic numbers in clusters

Abstract: For integers n ≥ 2 and Q> 1, we introduce the class of integer polynomials P (x )= anx n + an−1x n−1 + ··· + a1x + a0 such that

Torsion homology of arithmetic lattices and K2 of imaginary fields

Abstract: We study upper bounds for the torsion in homology of nonuniform arithmetic lattices. Together with recent results of Calegari-Venkatesh, this can be used to obtain upper bounds on K2 of the ring of integers of totally imaginary fields.