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A Subexponential Algorithm for the Determination of Class Groups and Regulators of Algebraic Number Fields
, 1990
"... A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , wh ..."
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Cited by 51 (5 self)
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A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , where c 2 R?0 is an absolute constant, and where the o(1)function depends on the degree of F . 1 Introduction Computing the class group and the regulator of an algebraic number field F are two major tasks of algorithmic algebraic number theory. In the last decade, several regulator and class group algorithms have been suggested (e.g. [16],[17],[18],[3]). In [2] the problem of the computational complexity of those algorithms was adressed for the first time. This question was then studied in [2] in great detail. The theoretical results and the computational experience show that computing class groups and regulators is a very difficult problem. More precisely, it turns out that even under the a...
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 40 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Algorithmic enumeration of ideal classes for quaternion orders
 SIAM J. Comput. (SICOMP
"... Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal ..."
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Cited by 12 (7 self)
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Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2. Key words. quaternion algebras, maximal orders, ideal classes, number theory AMS subject classifications. 11R52 Since the very first calculations of Gauss for imaginary quadratic fields, the problem of computing the class group of a number field F has seen broad interest. Due to the evident close association between the class number and regulator (embodied in the Dirichlet class number formula), one often computes the class group and unit group in tandem as follows. Problem (ClassUnitGroup(ZF)). Given the ring of integers ZF of a number field F, compute the class group Cl ZF and unit group Z ∗ F.
Computing Arakelov class groups
, 2008
"... Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of ..."
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Cited by 7 (0 self)
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Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.
Approximating Euler products and class number computation in algebraic function fields
"... Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suit ..."
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Cited by 3 (3 self)
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Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1.
Distributed Class Group Computation
 Informatik  Festschrift aus Anla des sechzigsten Geburtstages von Herrn Prof. Dr. G. Hotz, volume 1 of TeubnerTexte zur Informatik
, 1991
"... We present an improved sequential and a parallel version of the algorithm of Hafner and McCurley for the computation of the class group of imaginary quadratic fields. We describe the implementation of this algorithm on a network of UNIXworkstations using the system LIPS. ..."
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Cited by 2 (1 self)
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We present an improved sequential and a parallel version of the algorithm of Hafner and McCurley for the computation of the class group of imaginary quadratic fields. We describe the implementation of this algorithm on a network of UNIXworkstations using the system LIPS.
Baby Step Giant Step in Real Quadratic Function Fields
, 1995
"... The principal topic of this article is to extend Shanks' infrastructure ideas in real quadratic number fields to the case of real quadratic congruence function fields. We apply these techniques to the problem of computing the regulator R of a real quadratic congruence function field. By making use o ..."
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Cited by 2 (1 self)
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The principal topic of this article is to extend Shanks' infrastructure ideas in real quadratic number fields to the case of real quadratic congruence function fields. We apply these techniques to the problem of computing the regulator R of a real quadratic congruence function field. By making use of symmetries and optimized formulas we obtain a considerable improvement in the complexity of calculating R. We also apply the ideas of Lenstra and Schoof to produce an algorithm for determining the regulator unconditionally in O(q 1 5 deg(D)+" ) operations. 1 Introduction Let k = F q be a finite field of odd characteristic with q elements and let K = k(x)( p D), where D is a monic, squarefree polynomial of even degree. Such a field is known as a real quadratic congruence function field (of odd characteristic). Note that K is a Galois extension of the rational function field k(x) with Galois group f1; oeg, where oe is the Kautomorphism which takes p D to \Gamma p D. The conjugate o...
Approximate Evaluation of L(1, χ_Δ)
, 1998
"... . We develop a framework for computing with rational approximations to real numbers in number theoretic computations. We use that framework to analyze the approximate evaluation of the value L(1; \Delta ) where L(s; \Delta ) is the Lfunction of the quadratic order of discriminant \Delta. 1. Intr ..."
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. We develop a framework for computing with rational approximations to real numbers in number theoretic computations. We use that framework to analyze the approximate evaluation of the value L(1; \Delta ) where L(s; \Delta ) is the Lfunction of the quadratic order of discriminant \Delta. 1. Introduction In many computations in number theory approximations to real numbers are used. Important examples are the computation of values of Lseries, in particular the approximation of the product of the regulator and the class number of a number field by means of the analytic class number formula (see for example [Sha72], [Len82], [BW89], [Coh95], [JLW95]) and the determination of a system of fundamental units and the regulator from a generating set of units (see for example [Buc89],[PZ89], [Coh95]). Those computations are carried out with rational approximations of a certain precision. Therefore, roundoff errors occur and those errors have to be taken into account. Unfortunately, most alg...
Computing Arakelov class groups
, 801
"... Abstract. Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class grou ..."
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Abstract. Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context. 1. Introduction. In his 1972 Boulder paper [26], Daniel Shanks observed that the quadratic forms in the principal cycle of reduced binary quadratic forms of positive discriminant exhibit a grouplike behavior. This was a surprising phenomenon, because the principal cycle itself constitutes the trivial class of the class group. Shanks called this grouplike structure ‘inside’