Showing papers on "Ring of integers published in 2007"

Lattices that admit logarithmic worst-case to average-case connection factors

Abstract: We exhibit an average-case problem that is as hard as finding γ(n)-approximate shortest nonzero vectors in certain n-dimensional lattices in the worst case, for γ(n) = O(√log n). The previously best known factor for any non-trivial class of lattices was γ(n) = O(n).Our results apply to families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case problem we rely on is to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field.For the connection factors γ(n) we achieve, the corresponding decision problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices.To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3-e) would yield connection factors better than O(n).As an additional contribution, we give reductions between various worst-case problems on ideal lattices, showing for example that the shortest vector problem is no harder than the closest vector problem. These results are analogous to previously-known reductions for general lattices.

Non-additive geometry

Abstract: We develop a language that makes the analogy between geometry and arithmetic more transparent. In this language there exists a base field $\mathbb{F}$ , ‘the field with one element’; there is a fully faithful functor from commutative rings to $\mathbb{F}$ -rings; there is the notion of the $\mathbb{F}$ -ring of integers of a real or complex prime of a number field $K$ analogous to the $p$ -adic integers, and there is a compactification of $\operatorname{Spec}O_K$ ; there is a notion of tensor product of $\mathbb{F}$ -rings giving the product of $\mathbb{F}$ -schemes; in particular there is the arithmetical surface $\operatorname{Spec} O_K\times\operatorname{Spec} O_K$ , the product taken over $\mathbb{F}$ .

Computing endomorphism rings of Jacobians of genus 2 curves over finite fields

Abstract: We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups $J[\ell^d]$ for prime powers $\ell^d$. We use these algorithms to create the first implementation of Eisentrager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute $p^3$ curves for many small primes $p$.

Units generating the ring of integers of complex cubic fields

Abstract: All purely cubic fields such that their maximal order is generated by its units are determined.

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

Abstract: We define and investigate extension groups in the context of Arakelov geometry. The 'arithmetic extension groups' we introduce are extensions by groups of analytic types of the usual extension groups attached to $\O_X$-modules over an arithmetic scheme $X$. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over $X$ - and we especially consider the case when $X$ is an 'arithmetic curve', namely the spectrum $\Spec \O_K$ of the ring of integers in some number field $K$. Then the study of arithmetic extensions over $X$ is related to old and new problems concerning lattices and the geometry of numbers.

ANALYTIC PROPERTIES OF THE q-VOLKENBORN INTEGRAL ON THE RING OF p-ADIC INTEGERS

Abstract: In this paper, we consider the q-Volkenborn integral of uniformly differentiable functions on the p-adic integer ring By using this integral, we obtain the generating functions of twisted q-generalized Bernoulli numbers and polynomials We find some properties of these numbers and polynomials

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Abstract: Let K = Q(√-l) be an imaginary quadratic field with ring of integers O K, where l is a square free integer such that l ≡ 3 mod 4, and let C = [n, k] is a linear code defined over O k /2O k . The level I theta function Θ Λl (C) of C is defined on the lattice Λ l (C):= {x ∈ O n k : ρ l (x) ∈ C}, where ρl O K → O K /20 K is the natural projection. In this paper, we prove that: i) for any l,l' such that l≤l', ⊖Λ l (q) and ⊖Λ l' (q) have the same coefficients up to q l+1/4, ii) for l ≥ 2(n+1)(n+2)/n -1, ⊖Λ l (C) determines the code C uniquely, iii) for l < 2(n+1)(n+2)/n - 1, there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to ⊖ Λl (C).

On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings

Abstract: We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_\lambda$ of torsion $\Oh$ obreakdash-modules. When $n$ is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of $\GL_n(F)$. We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of $\GL_n(\Oh_k)$ for $k\geq 2$ for all $n$ is equivalent to the construction of the representations of all the groups $G_\lambda$. A functional equation for zeta functions for representations of $\GL_n(\Oh_k)$ is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ are constructed. Not all these representations are strongly cuspidal.

Local galois module structure in positive characteristic and continued fractions

Abstract: For a Galois extension of degree p of local fields of characteristic p, we express the Galois action on the ring of integers in terms of a combinatorial object: a balanced {0, 1}-valued sequence that only depends on the discriminant and p. We show that the embedding dimension edim(R) of the associated order R is tightly related to the minimal number d of R-module generators of the ring of integers. Moreover, we show how to compute d and edim(R) from p and the discriminant with a continued fraction expansion.

Hecke operators and Hilbert modular forms

Abstract: Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms.

Surfaces over a p-adic field with infinite torsion in the Chow group of 0-cycles

Abstract: We give an example of a projective smooth surface X over a p-adic field K such that for any prime ` different from p, the `-primary torsion subgroup of CH0(X), the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving a variant of the Bloch-Kato conjecture which characterizes the image of an `-adic regulator map from a higher Chow group to a continuous etale cohomology of X by using p-adic Hodge theory. With the aid of the theory of mixed Hodge modules, we reduce the problem to showing the exactness of the de Rham complex associated to a variation of Hodge structure, which is proved by the infinitesimal method in Hodge theory. Another key ingredient is the injectivity result on the cycle class map for Chow group of 1-cycles on a proper smooth model of X over the ring of integers in K, due to K. Sato and the second author.

On the restriction of representations of GL2(F) to a Borel subgroup

Abstract: Let F be a non-Archimedean local field and let p be the residual characteristic of F. Let G=GL2(F) and let P be a Borel subgroup of G. In this paper we study the restriction of irreducible smooth representations of G on -vector spaces to P. We show that in a certain sense P controls the representation theory of G. We then extend our results to smooth -modules of finite length and unitary K-Banach space representations of G, where is the ring of integers of a complete discretely valued field K with residue field .

Permutation polynomial based interleavers for turbo codes over integer rings: theory and applications

Abstract: Turbo codes are a class of high performance error correcting codes (ECC) and an interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions of interleavers are of particular interest because they admit analytical designs and simple, practical hardware implementation. Sun and Takeshita [33] have shown that the class of quadratic permutation polynomials over integer rings provides excellent performance for turbo codes. Recently, quadratic permutation polynomial (QPP) based interleavers have been proposed into 3rd Generation Partnership Project Long Term Evolution (3GPP LTE) draft [55] for their excellent error performance, simple implementation and algebraic properties which admit parallel processing and regularity. In some applications, such as deep space communications, a simple implementation of deinterleaver is also of importance. In this dissertation, a necessary and sufficient condition is proven for the existence of a quadratic inverse polynomial (deinterleaver) for a quadratic permutation polynomial over an integer ring. Further, a simple construction is given for the quadratic inverse. We also consider the inverses of QPPs which do not admit quadratic inverses. It is shown that most 3GPP LTE interleavers admit quadratic inverses. However, it is shown that even when the 3GPP LTE interleavers do not admit quadratic inverses,

Remarks on polynomial parametrization of sets of integer points

Abstract: If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show that every co-finite subset of Z^k is parametrizable a single k-tuple of polynomials with integer coefficients.

Computing endomorphism rings of Jacobians of genus 2 curves over finite fields.

Abstract: We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups $J[\ell^d]$ for prime powers $\ell^d$. We use these algorithms to create the first implementation of Eisentrager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute $p^3$ curves for many small primes $p$.

Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field

Abstract: W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.

The generalized artin conjecture and arithmetic orbifolds

Abstract: Let K be a number field with positive unit rank, and let OK denote the ring of integers of K. A generalization of Artin’s primitive root conjecture is that that OK is a primitive root set for infinitely many prime ideals. We prove this with additional conjugacy conditions in the case when K is Galois with unit rank greater than three. This was previously known under the assumption of the Generalized Riemann Hypothesis. From our result, we deduce a topological corollary about the structure of quotients of PSL2(OK).

On quasi-cyclic codes over integer residue rings

Abstract: In this paper we consider some properties of quasi-cyclic codes over the integer residue rings. A quasi-cyclic code over Zk, the ring of integers modulo k, reduces to a direct product of quasi-cyclic codes over Zpi ei, k = &Pi:i = 1s piei, pi, a prime. Let T be the standard shift operator. A linear code C over a ring R is called an l-quasi-cyclic code if Tl(c) ∈ C, whenever c ∈ C. It is shown that if (m, q) = 1, q = pr, p a prime, then an l-quasi-cyclic code of length lm over Zq is a direct product of quasi-cylcic codes over some Galois extension rings of Zq. We have discussed about the structure of the generator of a 1-generator l- quasi-cyclic code of length lm over Zq. A method to obtain quasi-cyclic codes over Zq, which are free modules over Zq, has been discussed.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

Abstract: Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$, and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

Galois actions on torsion points of universal one-dimensional formal modules

Abstract: Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show that the representation of the fundamental group of $U_h$ on the Tate module of the etale quotient is surjective.

Coniveau over $p$-adic fields and points over finite fields

Abstract: If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).

Abstract factorials

Abstract: A commutative semigroup of abstract factorials is defined in the context of the ring of integers We study such factorials for their own sake, whether they are or are not connected to sets of integers Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational

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.

Arithmetic of the module of roots of the isogeny of a formal group in the case of small ramification

Abstract: In this paper, for a one-dimensional formal group over the ring of integers of a local field in the case of small ramification we study the arithmetic of the module of roots of the isogeny, as well as the arithmetic of the formal module constructed on the maximal ideal of a local field containing all the roots of the isogeny. Bibliography: 5 titles.

Ihara's lemma for imaginary quadratic fields

Abstract: An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal P of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard P-degeneracy maps between the cuspidal sheaf cohomology H^1_!(X_0, M_0)^2 --> H^1_!(X_1, M_1) is Eisenstein. Here X_0 and X_1 are analogues over F of the modular curves X_0(N) and X_0(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL(2) which is due to Serre.