Showing papers on "Ring of integers published in 2009"

Characterizing integers among rational numbers with a universal-existential formula

Abstract: We prove that ${\Bbb Z}$ in definable in ${\Bbb Q}$ by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of ${\Bbb Q}$-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.

Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$

Abstract: We introduce a new notion of modular independence to define bases and the generator matrices for the codes over the ring of integers $${\mathbb {Z}_m}$$ of general modulus m. We define standard forms for such generator matrices, and discuss how to find such forms and the parity check matrices.

Computations in Relative Algebraic K-Groups

Abstract: Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

On the Coates–Sinnott Conjecture

Abstract: In [5], Coates and Sinnott formulated a far reaching conjecture linking the values ΘF/k,S(1 — n) for even integers n ≥ 2 of an S -imprimitive, Galois-equivariant L -function ΘF/k,S associated to an abelian extension F/k of totally real number fields to the annihilators over the group ring ℤ[G (F/k)] of the even Quillen K -groups K2n–2 (OF) associated to the ring of integers OF of the top field F. In the same paper, Coates and Sinnott essentially prove the -adic etale cohomological version of their conjecture, in which K2n–2(OF) is replaced by H2et(OF [1/ ], ℤ(n)), for all primes > 2, under the hypothesis that k = ℚ. Refinements of this result for k = ℚ, involving Fitting ideals rather than annihilators of H2et(OF [1/], ℤ(n)), were obtained in particular cases by Cornacchia–Ostvaer [7] and in general by Kurihara [14]. More recently, Burns and Greither [3] proved the same type of refinements (involving Fitting ideals of etale cohomology groups) for arbitrary totally real base fields k, but working under the very strong hypothesis that the Iwasawa μ -invariants μF, vanish for all odd primes . In this paper, we study a class of abelian extensions of an arbitrary totally real base field k including, for example, subextensions of real cyclotomic extensions of type k (ζ)+/k, where p is an odd prime. For this class of extensions, we prove similar refinements of the etale cohomological version of the Coates–Sinnott conjecture, under no vanishing hypotheses for the Iwasawa μ-invariants in question. Our methods of proof are different from the ones employed in [3], [14] and [7]. We build upon ideas developed by Greither in [10] and Wiles in [23] and [22], in the context of Brumer's Conjecture. If the Quillen–Lichtenbaum Conjecture is proved (and a proof seems tobe within reach), then we have canonical ℤ[G (F/k)]-module isomorphisms for all n ≥ 2, all i = 1,2, and all primes > 2, and all these results will yield proofs of the original K -theoretic version of the Coates–Sinnott Conjecture, in the cases and under the various hypotheses mentioned above (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Endomorphisms of superelliptic jacobians

Abstract: Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.

Elementary number theory : primes, congruences, and secrets : a computational approach

Abstract: Prime Numbers.- The Ring of Integers Modulo n.- Public-key Cryptography.- Quadratic Reciprocity.- Continued Fractions.- Elliptic Curves.

On the Analysis of Cryptographic Assumptions in the Generic Ring Model

Abstract: At Eurocrypt 2009 Aggarwal and Maurer proved that breaking RSA is equivalent to factoring in the generic ring model . This model captures algorithms that may exploit the full algebraic structure of the ring of integers modulo n , but no properties of the given representation of ring elements. This interesting result raises the question how to interpret proofs in the generic ring model. For instance, one may be tempted to deduce that a proof in the generic model gives some evidence that solving the considered problem is also hard in a general model of computation. But is this reasonable? We prove that computing the Jacobi symbol is equivalent to factoring in the generic ring model. Since there are simple and efficient non-generic algorithms computing the Jacobi symbol, we show that the generic model cannot give any evidence towards the hardness of a computational problem. Despite this negative result, we also argue why proofs in the generic ring model are still interesting, and show that solving the quadratic residuosity and subgroup decision problems is generically equivalent to factoring.

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.

Quaternary universal forms over $\mathbf{Q}(\sqrt{13})$

Abstract: Let \(F=\mathbf{Q}(\sqrt{m})\) be a real quadratic field over Q with m a square-free positive rational integer and \(\mathcal{O}\) be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x1,…,xn)=∑1≤i,j≤nαijxixj ( \(\alpha_{ij}=\alpha_{ji}\in \mathcal{O}\) ) is called universal if f represents all totally positive integers in \(\mathcal{O}\) . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12.

Arithmetic Intersection Theory on Deligne-Mumford Stacks

Abstract: The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field

Extended Deligne-Lusztig varieties for general and special linear groups

Abstract: We give a generalisation of Deligne-Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne-Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne-Lusztig varieties, and show that the "unramified" varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for $\SL_{2}(\mathbb{F}_{q}[[\varpi]]/(\varpi^{2}))$, with odd $q$, the extended Deligne-Lusztig varieties do indeed afford all the irreducible representations.

Density results for automorphic forms on Hilbert modular groups II

Abstract: We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips and products of prescribed small intervals for all but one of the infinite places of $F$. The main tool in the derivation is a sum formula of Kuznetsov type.

The Tame Kernel of Multiquadratic Number Fields

Abstract: Let F/K be a Galois extension of a number field of degree n, 𝒪 F the ring of integers in F, and p a prime number which does not divide n. Let K 2 denote the Milnor K-functor. In this article, we shall study the structure of the odd part of the tame kernel K 2𝒪 F of F by using the intermediate fields of F/K. In particular, for a multiquadratic field F, we shall get the p i -rank, (i > 0) of K 2𝒪 F . Finally, we shall determine the structure of the odd parts of K 2𝒪 F when where − 100 < d < 0, d 1 = 2,3,5,7.

Reduction modulo p of cuspidal representations and weights in Serre's conjecture

Abstract: Let O be the ring of integers of a p-adic field and p its maximal ideal. We compute the Jordan-Holder decomposition of the reduction modulo p of the cuspidal representations of GL2(O/p) for e ≥ 1. We also provide an alternative formulation of Serre’s conjecture for Hilbert modular forms. 1. Cuspidal representations and weights 1.1. Cuspidal representations. Let K/Qp be a local field, where p is a prime, and let O be the ring of integers and p its maximal ideal. Let Re = O/pe. In particular, R1 = O/p is the residue field; let q = pf be its cardinality. Let K be the unramified quadratic extension of K, and let O and p be its ring of integers and maximal ideal. The cuspidal complex representations of GL2(Re) are well known (see for instance [PS]) in the case e = 1 and have been constructed for general e, under various names, by several authors; see, for instance, [Shi], [Ger], [How], [Car], [BK], and [Hil]. Aubert, Onn, and Prasad proved ([AOP], Theorem B; note that the notions of cuspidal and strongly cuspidal representations coincide for GL2 by Theorem A) that they are parametrized by Gal(K/K)-orbits of strongly primitive characters ξ : (O/pe)∗ → C∗. A strongly primitive character of (O/p)∗ is one that does not factor through the norm map N : O/p → O/p. See [AOP], 5.2, for the definition of strongly primitive characters for general e. We denote by Θe(ξ) the cuspidal representation of GL2(Re) corresponding to ξ. Fix an isomorphism C ' Qp, and from now on we view ξ and Θe(ξ) as p-adic representations. In this note we compute the Jordan-Holder constituents of Θe(ξ), the reduction mod p of Θe(ξ), and use the notions introduced to reformulate the Serre-type conjecture for Hilbert modular forms of [Sch]. See the last section for some remarks about motivation. The author is very grateful to the referee for comments that improved the exposition, and particularly for an observation that considerably simplified the computations in section 2. 1.2. Brauer characters. Let Θe(ξ) be a cuspidal representation of GL2(Re). The Jordan-Holder constituents of Θe(ξ) are determined by its Brauer character, hence by the values of the character of Θe(ξ) at p-regular conjugacy classes. The p-regular conjugacy classes of GL2(Re) are sent by the natural surjection π : GL2(Re) → GL2(R1) to p-regular conjugacy classes of GL2(R1). Moreover, Date: May 24, 2008. 1

Sur la classification de quelques φ-modules simples (Appendice à On the structure of some moduli spaces of finite flat group schemes par E. Hellmann)

Abstract: This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (b > 1 is an integer) and acting on elements of k through a fixed automorphism. Some of these phi-modules are involved in the classification of finite flat group schemes over ring of integers of p-adic fields.

Local conditions for global representations of quadratic forms

Abstract: We show that the theorem of Ellenberg and Venkatesh on rep- resentation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form. In the article (5) Ellenberg and Venkatesh prove that for any integral positive definite quadratic formf inn variables there is a constantC(f) such that integral quadratic forms g of square free discriminant in m ≤ n − 5 variables with minimum µ(g) > C(f) are represented byf if and only if they are represented byf locally everywhere, i.e., over R and over all p-adic integers. If one fixes an odd prime p not dividing the discriminant of f one can find a constant C ' (f,p) such that representability is even guaranteed for g of rank m ≤ n − 3 with µ(g) > C ' (f,p), provided the discriminant of g is further restricted to be prime to p. It is mentioned in (5) that I have suggested to replace the condition of square free discriminant on g by a weaker condition. This suggestion is worked out here. Combining our version of the result of (5) with results of Kitaoka we also obtain some new cases in which with a suitable fixed prime q the only condition on g (apart from µ(g) > C(f,q) and representability of g by f locally everywhere) is bounded divisibility of the discriminant of g by q. Moreover, results on extensions of representations as given in (1, 2) can be obtained with new dimension bounds. We take the occasion to reformulate some of the proofs of (5) in a way that is closer to other work on the subject. We will work throughout in the language of lattices as described e.g. in (12, 15) (with the exception of Theorem 11). We fix a totally real number field F with ring of integers o and a totally positive definite quadratic space (V,Q) overF of dimension n ≥ 3; the quadratic form Q may be written as Q(x) = h x,xi with a scalar product h , i on V. By OV (F) we denote the group of isometries of V with respect to Q (the orthogonal group of the quadratic space (V,Q)), by OV (A) its adelization, by SOV (F) resp. SOV (A) their subgroups of elements of determinant 1. For a lattice � onV we denote its automorphism group (or unit group) {� ∈ OV (F) | �(�) = �} by O�(o), similarly for the local or adelic analogues. The minimum ofis

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

Abstract: All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.

An algorithm for constructing the basis of the solution set for systems of linear Diophantine equations over the ring of integers

Abstract: A polynomial algorithm is proposed to construct the minimal generating set of solutions and the basis of the solution set for systems of linear Diophantine equations over the ring of integer. The algorithm is based on a modified TSS method.

A finer classification of the unit sum number of the ring of integers of quadratic fields and complex cubic fields

Abstract: The unit sum number, u(R), of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending on whether the units generate R additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is ω or ∞. Then we present some examples showing that all possibilities can occur.

Pure subrings of the rings \mathbb Z_\chi

Abstract: Pure subrings of finite rank in the -adic completion of the ring of integers and in its homomorphic images are considered. Certain properties of these rings are studied (existence of an identity element, decomposability into a direct sum of essentially indecomposable ideals, condition for embeddability into a csp-ring, etc.). Additive groups of these rings and conditions under which these rings are subrings of algebraic number fields are described. Bibliography: 12 titles.

On the Number of Equivalence Classes of Attracting Dynamical Systems

Abstract: We study discrete dynamical systems of the kind h(x) = x + g(x), where g(x) is a monic irreducible polynomial with coefficients in the ring of integers of a p-adic field K. The dynamical systems of ...

On the Belyi degree(s) of a curve defined over a number field

Abstract: Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree of such a cover. More precisely, we start by defining the absolute Belyi degree of X, which only depends on the $\bar{\bold Q}$-isomorphism class of $X$. We then give a lower bound of this invariant, only depending on the stable primes of bad reduction (as defined in the paper) and we show that this bound is sharp. In the second part of the paper, we introduce the relative Belyi degree of a curve X defined over a fixed number field $K$. We first prove that there exist finitely many $K$-isomorphism classes of curves of bounded (relative) Belyi degree and we then obtain a lower bound, only depending on the primes of bad reduction of the minimal regular model of $X$ over (the ring of integers of) $K$.