Algorithmic Algebra and Number Theory: Selected Papers From a Conference Held at the University of Heidelberg in October 1997 PDF
Preview Algorithmic Algebra and Number Theory: Selected Papers From a Conference Held at the University of Heidelberg in October 1997
Matzat· Greuel· Hiss (Eds.) Algorithmic Algebra and Number Theory Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo B. Heinrich Matzat Gert -Martin Greuel Gerhard Hiss (Eds.) Algorithmic Algebra and Number Theory Selected Papers from a Conference Held at the University of Heidelberg in October 1997 t Springer B. Heinrich Matzat Interdisziplinares Zentrum fUr Wissenschaftliches Rechnen der Universitat Heidelberg 1m Neuenheimer Feld 368 D-69120 Heidelberg, Germany e-mail: [email protected] Gert-Martin Greuel Fachbereich Mathematik Universitat Kaiserslautern Postfach 3049 D-67653 Kaiserslautern Germany e-mail: [email protected] Gerhard Hiss RWTHAachen Lehrstuhl D fUr Mathematik Templergraben 64 D-52062 Aachen, Germany e-mail: [email protected] Mathematics Subject Classification (1991): 11-06,12-06,13-06,14-06,20-06, nY4 0, 12 Y0 5, 13PI0, 14QXX, 20B40, 20C40, 68Q40 Cataloging-in-Publication Data applied for Algorithmic algebra and number theory: selected papers from a conference, held at the University of Heidelberg in October 1997 / B. Heinrich Matzat ... (ed.).-Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 ISBN-13: 978-3-540-64670-9 e-ISBN-13: 978-3-642-59932-3 DOl: 10.1007/978-3-642-59932-3 ISBN-13: 978-3-540-64670-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting. reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: By the authors using a Springer LaTeX Macro Package Cover design: design & production GmbH, Heidelberg SPIN: 10654649 46/3143 -5 4 3 2 1 0 - Printed on acid-free paper Preface This book contains 22 lectures presented at the final conference of the Ger man research program (Schwerpunktprogramm) Algorithmic Number The ory and Algebra 1991-1997, sponsored by the Deutsche Forschungsgemein schaft. The purpose of this research program and of the meeting was to bring together developers of computer algebra software and researchers using com putational methods to gain insight into experimental problems and theoret ical questions in algebra and number theory. The book gives an overview on algorithmic methods and on results ob tained during this period. This includes survey articles on the main research projects within the program: • algorithmic number theory emphasizing class field theory, constructive Galois theory, computational aspects of modular forms and of Drinfeld modules • computational algebraic geometry including real quantifier elimination and real algebraic geometry, and invariant theory of finite groups • computational aspects of presentations and representations of groups, especially finite groups of Lie type and their Heeke algebras, and of the isomorphism problem in group theory. Some of the articles illustrate the current state of computer algebra sys tems and program packages developed with support by the research pro gram, such as KANT and LiDIA for algebraic number theory, SINGULAR, RED LOG and INVAR for commutative algebra and invariant theory respec tively, and GAP, SYSYPHOS and CHEVIE for group theory and representation theory. According to the three main research directions, the book is divided into three parts representing algorithmic aspects of algebraic number the ory, commutative algebra and algebraic geometry, and group theory and representation theory, edited by B. H. Matzat, G.-M. Greuel and G. Hiss, respectively. The editors thank the contributors to this volume and the Deutsche Forschungsgemeinschaft for its support of the research program and the conference held in Heidelberg. G.-M. Greuel, G. Hiss and B. H. Matzat Heidelberg, May 1998 Table of Contents Part A: Algorithmic Algebraic Number Theory Sieving Methods for Class Group Computation J. Buchmann, M. J. Jacobson, Jr., S. Neis, P. Theobald and D. Weber ...... 3 Arithmetic of Modular Curves and Applications G. Prey and M. Muller.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 Local and Global Ramification Properties of Elliptic Curves in Characteristics Two and Three E. -U. Gekeler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49 Techniques for the Computation of Galois Groups A. Hulpke ................................................................. 65 Fortschritte in der inversen Galoistheorie B. H. Matzat .............................................................. 79 From Class Groups to Class Fields M. E. Pohst ............................................................. 103 A Gross-Zagier Formula for Function Fields H.-G. Ruck and U. Tipp ................................................. 121 Extremal Lattices R. Scharlau and R. Schulze-Pillot ......................................... 139 Part B: Algorithmic Commutative Algebra and Algebraic Geometry On the Real Nullstellensatz E. Becker and J. Schmid ................................................. 173 Primary Decomposition: Algorithms and Comparisons W. Decker, G.-M. Greuel and G. Pfister .................................. 187 Real Quantifier Elimination in Practice A. Dolzmann, T. Sturm and V. Weispfenning ............................ 221 Hilbert Series and Degree Bounds in Invariant Theory G. Kemper ............................................................... 249 Invariant Rings and Fields of Finite Groups G. Kemper and G. Malle ................................................. 265 Computing Versal Deformations with SINGULAR B. Martin ................................................................ 283 VIII Table of Contents Algorithms for the Computation of Free Resolutions T. Siebert ................................................................ 295 Part C: Algorithmic Group and Representation Theory Computational Aspects of the Isomorphism Problem F. M. Bleher, W. Kimmerle, K. W. Roggenkamp and M. Wursthorn ...... 313 Representations of Hecke Algebras and Finite Groups of Lie Type R. Dipper, M. Geck, G. Hiss and G. Malle ............................... 331 The Groups of Order 512 B. Eick and E. A. O'Brien ............................................... 379 Computational Aspects of Representation Theory of Finite Groups II K. Lux and H. Pahlings .................................................. 381 High Performance Computations in Group Representation Theory G. O. Michler ............................................................ 399 The Structure of Maximal Finite Primitive Matrix Groups G. Nebe .................................................................. 417 Presentations and Representations of Groups W. Plesken .............................................................. 423 Part A Algorithmic Algebraic N umber Theory Sieving Methods for Class Group Computation Johannes Buchmann and Michael J. Jacobson, Jr. and Stefan Neis and Patrick Theobald and Damian Weber Institut fiir Theoretische Informatik Technische Universitat Darmstadt 1 Introduction Computing the class group and regulator of an algebraic number field K are two major tasks of algorithmic algebraic number theory. The asymptotically fastest method known has conjectured sub-exponential running time and was proposed in [5]. In this paper we show how sieving methods developed for factoring algo rithms can be used to speed up this algorithm in practice. We present numerical experiments which demonstrate the efficiency of our new strategy. For example, we are able to compute the class group of an imaginary quadratic field with a dis criminant of 55 digits 20 times as fast as S. Dtillmann in an earlier record-setting implementation ([1]) which did not use sieving techniques. We also present class numbers of large cubic fields. 2 The Algorithm We will consider the problem of computing the class group Cl(K) of an algebraic number field K given by an irreducible monic polynomial of degree n = r + 2s, where r is the number of real embeddings and s is the number of complex embeddings of K into the field <C of complex numbers. We denote the maximal order of K by OK. The norm of an algebraic number a will be denoted by N (a), and the norm of an ideal a will be denoted by N(a). The class number of K will be denoted by h and the regulator by R. We briefly review the algorithm presented in [5]. Let FB be a set of prime ideals over K and k = WBI. For e = (e1, ... ,ek) E 2Zk, we write IkI = FBe p~i. i=l By AFB we denote all algebraic numbers a in K which, considered as principal ideals, can be represented as a power product of the ideals of the factor base FB, i.e., AFB = {aE K I a·OK =FBe,eE 2Zk}. Consider the maps B. H. Matzat et al. (eds.), Algorithmic Algebra and Number Theory © Springer-Verlag Berlin Heidelberg 1999 4 J. Buchmann, M.J. Jacobson Jr., S. Neis, P. Theobald and D. Weber and P: AFB ~ ZZk X IRr+s- I a I---t (e, log lUI (a) I, ... , log IUr+8-I (a)l), where a· OK = FEe and the Ui are the embeddings of K into <C. By [5, Theorem 2.1] we know: Theorem 1. Suppose that the ideal classes of the elements of FE generate the class group of K. Then P(AFB) is a (k + r + s - I)-dimensional lattice with determinant hR. Also, pi (AF B) is a k-dimensional lattice with determinant h and we have ZZk/pl(AFB) ~ Cl(K). Based on this theorem, we can compute the class number and regulator by finding relations, Le., algebraic numbers a together with their decompositions over the factor base. Having found enough relations to generate a sub-lattice of p( AFB ) offull dimension, it can be checked whether these relations generate the full lattice by applying the analytic class number formula 2r(21f)s IT 1 - !. -w--y';r: =I."=1-'' =K=II h R = pprime IIl-:pN I'( Pj ' pip where .1K denotes the discriminant of the field K and w denotes the number of roots of unity in K. This formula enables us to compute a number h* with h* < hR < 2h*. If the determinant of the sublattice of P(AFB) is in this interval then the sub lattice is the full lattice. Otherwise, additional relations can be generated until we find the full lattice. 3 Generating Relations Several methods for finding relations among the factor base elements have been suggested, for example, in [1], [14], and [6]. We show how to use sieving techniques known from factoring algorithms to compute relations very efficiently. 3.1 Quadratic Fields For quadratic fields, we can apply a modification of the multiple polynomial quadratic sieve (for details, see [13]). Suppose that we find a E K, v, e E ZZk such that (a)/ FEU = FEe. If K is imaginary quadratic then (e+v) E pl(AFB). If K is real quadratic then (e + v, log lUI (a) I) E P(AFB)' To find such numbers a, we proceed as follows: - Pick a random exponent v E {-I, O,I}k and compute the ideal a = FEu. Let a = aZZ+ btf1ZZ. Then, all elements in a have norm af(x, V), x, Y E ZZ, where f = aX2 + bXY + cy2 E ZZ[X, Y] and c = (b2 - .1)/(4a).
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