Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

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.

On double cyclic codes over Z_4

Abstract: Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

Rings of continuous integer-valued functions and nonstandard arithmetic

Abstract: 0. Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains. These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest. Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered. In ?1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to 6X (1.2), the Boolean space of the algebra of open-and-closed sets of X. The maximal ideal space of C(X, Z) is shown to map continuously onto 6X (1.3). The space, 6OX, of points of 3X that give rise to integer residue class domains, is studied in ?2. The map of X into boX strongly resembles the realcompactification injection [GJ]. A representation theorem of C(X, Z) over 3OX is also given (2.4). It is shown in ?3 that points in 3X 6OX give rise to Z, a nonstandard model of Z (3.1). Here some of the relevant background material in model theory is discussed. The algebraic theory of nonstandard arithmetic is studied in ?4. In ?5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q. In ?6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3). The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal. The ideals of Z are analyzed in ?7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4). Ideals in C(X, Z) are decomposed in ?8, first into coprimary ideals (8.4), and then into primary ideals (8.9). In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X, Z) are analyzed. In ?9 some model-theoretic results are obtained on the residue class fields of C(X, Z), the principal result being that any such field is elementarily equivalent

Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

Abstract: It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field K. This includes the case where K is a p-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjolin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of th...

The smooth representations of GL_2(o).

Abstract: We give a classification of the smooth (complex) representations of GL2(𝔬), where 𝔬 is the ring of integers in a non-Archimedean local field. The approach is based on Clifford theory of finite groups and a corresponding study of orbits and stabilizers. In terms of this classification, we identify the representations which are geometrically or infinitesimally induced, respectively.