Showing papers on "Ring of integers published in 1995"

Subgroup growth and congruence subgroups

Abstract: Letk be a global field,O its ring of integers,G an almost simple, simply connected, connected algebraic subgroups ofGL m , defined overk and Γ=G(O) which is assumed to be infinite. Let σ n (Г) (resp. γ n (Г) be the number of all (resp. congruence) subgroups of index at mostn in Γ. We show:(a) If char(k)=0 then:

Link complements arising from arithmetic group actions

Abstract: Let be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space . Assume further that Γ\ℍ3 has finite hyperbolic volume. The quotient-space Γ\ℍ3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ3 is homeomorphic to the complement of a link in S3. We apply our procedure to subgroups of low index in , where is the ring of integers in . As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups so that Γ\ℍ3 is homeomorphic to the complement of a link in S3.

Corps biquadratiques monogènes

Abstract: Let\(K = \mathbb{Q}(\sqrt {dm} ,\sqrt {dn} )\) be a biquadratic number field (where d,m,n∈ℤ, are uniquely determined); we say that it is monogenic if its ring of integers OK is of the form ℤ[θ]. We show that K is monogenic if and only if the two following conditions are satisfied: (i) 2δm=2δn+4(2−δd) where δ=0 or 1 is defined by mn≡(−1)δ mod4; (ii) the equation (u2-v2)2(2δm)-(u2+v2)2(2δn)=±1 has solutions in ℤ.

Normal integral bases and the Spiegelungssatz of Scholz

Abstract: Introduction. In Hilbert’s Zahlbericht one can find the first global result on the Galois module structure of rings of integers ([H], Satz 132). Slightly extended it states that if N/Q is a tame extension with abelian Galois group ∆, then oN , the ring of integers in N , is free as a module over the group ring Z∆; moreover, an explicit canonical algebraic integer a such that oN = Z∆a can be given. The numbers a (δ ∈ ∆), the algebraic conjugates of a, are said to form a normal integral basis of the field extension N/Q. This result has been the starting point for a modern development, which has led to the deep result that the structure of oN as a module over Z∆, for arbitrary tame Galois extensions of number fields N/K with Galois group ∆, is determined in terms of the symplectic root numbers of N/K (see [F] and [T1]). Frohlich’s book [F] also contains a detailed introduction and a rather complete list of references. For a more recent survey on Galois module theory we refer to [Ca–Ch–F–T]. In the special case that ∆ is of odd order the result mentioned above gives that oN is a free Z∆-module, but the proof does not provide an explicit basis. If one considers the richer oK∆-module structure of oN rather than the Z∆-module structure alone, then oN is expected to be “usually” not even free if K 6= Q. Results of Taylor show that by modifying both oN and oK∆ one can sometimes—if K and N are certain ray class fields over imaginary quadratic number fields—achieve the “ideal” of free modules with explicit generators (see [C–T]). However, if one decides not to modify the original classical problem of the determination of the oK∆-module structure of oN , then a natural question is to what extent, for given K and ∆, the realization of ∆ as a Galois group of a tame extension N/K is determined by the ramification of N/K together with the structure of oN as an oK∆-module. This point of view is worked out in [B2]. In a sense the core of the question is how rare unramified extensions which possess a normal integral basis are. We mention in passing that this question is equivalent to a special case of a problem considered by Taylor in recent work, that of determining the kernel

The AICE-CRT and digital signal processing algorithms: the complex case

Abstract: The Chinese remainder theorem is a fundamental technique widely employed in digital signal processing for designing fast algorithms for computing convolutions. Classically, it has two versions. One is over a ring of integers and the second is over a ring of polynomials with coefficients defined over a field. In our previous papers, we developed an extension to this well-known theorem for the case of a ring of polynomials with coefficients defined over a finite ring of integers. The objective was to generalize number-theoretictransforms, which turn out to be a special case of this extension. This paper focuses on the extension of the Chinese remainder theorem for processing complex-valued integer sequences. Once again, the present work generalizes the complex-number-theoretic transforms. The impetus for this work is provided by the occurrence of complex integer sequences in digital signal processing and the desire to process them using exact arithmetic.

Representations of double coset hypergroups and induced representations

Abstract: The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair\(SL(2,\mathfrak{K})//SL(2,R)\) (where\(\mathfrak{K}\) are a local field andR its ring of integers) as a polynomial hypergroup on ℕ0 and characterize the (proper) subset of its dual consisting of extensible representations.

Faltings modular height and self-intersection of dualizing sheaf.

Abstract: Let K be a number field, O_K the ring of integers of K and X a stable curve over O_K of genus g >= 2. In this note, we will prove a strict inequality ( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)), where $K_{X/S}$ is the canonically metrized dualizing sheaf of X over S = Spec(O_K) and Height_{Fal}(J(X_K)) is the Faltings modular height of the Jacobian of X_K. As corollary, for any constant A, the set of all stable curves X over O_K with ( (K_{X/S})^2 / [K : Q] ) <= A is finite under the following equivalence. For stable curves X and Y, X is equivalent to Y if X is isomorphic to Y over O_{K'} for some finite extension field K' of K.

Primitive elements in integral bases

Abstract: Clearly, (1.1) implies (1.2). The question whether (1.1) is true was first raised by A. Fajardo Miron and H. W. Lenstra, Jr. [5, Sec. 8] when they were looking for ways to find equation orders of small index in a given ring of integers. Independently, H. Cohen observed in 1989 from numerical examples that (1.2) always seems to hold for fields of small degree. Cohen was looking for integral bases that are small in some sense, and one may wonder if these small basis elements can all lie in small number fields. In Section 2 we will show that (1.1) is true for Galois extensions of prime power degree, and for certain other classes of extensions as well. We will deduce (1.1) for all fields of degree less than 12. It is not hard to see that (1.1) is false for any normal field whose Galois group is dihedral of order 12. In Section 3 we prove the stronger result that (1.2) is false for such a field too. Notation. Throughout this paper, K ⊂ L denotes an extension of number fields with rings of integers A ⊂ B. We let SK(B) be the A-module generated by integers that are not primitive, i.e., generated by all x ∈ B with K(x) 6= L. The subfield integer index s(L/K) ∈ Z>0 ∪ {∞} is defined to be the index [B : SK(B)]. Note that (1.1) is equivalent to s(L/Q) 6= 1. For a finite group G and Z[G]-module M we let SG(M) be the additive subgroup ofM generated by elements ofM that are fixed by some non-trivial

Uniqueness of Dualities

Abstract: For a commutative ring R we discuss uniqueness of functorial dualities of the category L R of locally compact R-modules (i e involutive contravariant endofunctors of L R) Roeder proved that in case R is the ring of integers (i e for locally compact abelian groups) Pontryagin duality is the unique functorial duality It was conjectured by Iv Prodanov that in case R is an algebraic number ring such a uniqueness is available if and only if R is a principal ideal domain We prove this conjecture for real algebraic number rings and we show that Prodanov’s conjecture fails in case R is an order in an imaginary quadratic number field

Algorithms for Polycyclic-by-finite Matrix Groups: Preliminary Report

Abstract: Let R be the ring of integers or a number field. We present several algorithms for working with polycyclic-by-finite subgroups of GL(n,R). Let G be a subgroup of GL(n,R) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-finite. For polycyclic-by-finite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-finite, providing an alternative to the algorithm proposed by Beals ([Be1]) for this problem. Baumslag, Cannonito, Robinson and Segal prove that the problem of determining whether or not a finitely generated subgroup of GL(n,Z) is polycyclic-by-finite is decidable and that the problem of testing membership in a polycyclic-by-finite subgroup of GL(n,Z) is also decidable ([BCRS]). In this report we extend these results by describing algorithms which appear to be suitable for computer implementation. Experimentation is needed to determine the range of input for which they are practical. Our method is to first reduce each problem to the corresponding problem for triangularizable matrix groups. The reduction is an easy consequence of the result of Dixon ([Di1]) that subgroups of the p-congruence subgroup of GL(n,Z_p) are connected in the Zariski topology, where Z_p is the ring of p-adic integers for a prime p. We then prove a structure theorem for triangularizable matrix groups that allows us to decide whether or not a matrix group is triangularizable and to reduce the problem of testing membership in a polycyclic, triangularizable matrix group to the corresponding problems for finitely generated abelian matrix groups and for finitely generated unitriangular matrix groups. In the case of an abelian matrix group we can find a presentation for the group, and our membership test can be made constructive. For these results we rely heavily on the work of Ge ([Ge]) concerning algorithms for multiplicative subgroups of a number field. We also rely on an algorithm of Beals ([Be2]) to decide whether or not a triangularizable matrix group is polycyclic.

Self-intersection of dualizing sheaves of arithmetic surfaces with reducible fibers

Abstract: Let K be an algebraic number field and O_K the ring of integers of K Let f : X --> Spec(O_K) be a stable arithmetic surface over O_K of genus g >= 2 In this short note, we will prove that if f has a reducible geometric fiber, then the self intersection of dualizing sheaf of X with Arakelov metric is greater than or equal to log(2)/6(g-1)

An additive function on a ring of integers in the imaginary quadratic field Q(√d) with class-number one

Abstract: LetB α(a) be an additive function on a ring of integers in the quadratic number fieldQ(√d) given byB α(a) = Σ p|a *N α(p) with a fixedα > 0, where the asterisk means that the summation is over the non-associate prime divisorsp of an integera inQ(√d), N(a) is the norm ofa. In this paper we obtain the asymptotic formula of Σ N(a)≤x *B α(a) in the case where the class-number ofQ(√d) is one.

Determining the Monogeneity of a Quartic Number Field

Abstract: This note presents a method that determines all power integral bases of a quartic number field by solving Thue equations of degrees 3 and 4. To this end, projective representations of the ring of integers by graded complete intersections are studied and a criterion for monogeneity in terms of projective representations is derived.