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.

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InhaltsangabeTable of Contents/Inhaltsverzeichnis Chapter 1: Algorithmic Algebraic Number Theory J.Buchmann, M.J.Jacobson Jr., S.Neis, P.Theobald and D.Weber: Sieving Methods for Class Group Computation G. Frey and M. M"uller: Arithmetic of Modular Curves and Applications E.-U. Gekeler: Local and Global Ramification Properties of Elliptic Curves in Characteristics two and three A. Hulpke: Techniques for the Computation of Galois Groups B. H. Matzat: Fortschritte in der inversen Galoistheorie M. E. Pohst: From Class Groups to Class Fields H.-G. R"uck and U. Tipp: A. Gross-Zagier formula for function fields R. Scharlau and R. Schulze-Pillot: Extremal lattices Chapter 2: Algorithmic Commutative Algebra and Algebraic Geometry E. Becker and J. Schmid: On the Real Nullstellensatz W. Decker, G.-M. Greuel and G. Pfister: Primary Decompositions: Algorithms and Comparisons A. Dolzmann, Th. Sturm and V. Weispfenning: Real Quantifier Elimination in Practice G. Kemper: Hilbert Series and Degree Bounds in Invariant Theory G. Kemper and G. Malle: Invariant rings and fields of finite groups B. Martin: Computing Versal Deformations with SINGULAR Th. Siebert: Algorithms for the computation of free resolutions Chapter 3: Algorithmic Group and Representation Theory F. M. Bleher, W. Kimmerle, K. W. Roggenkamp and M. Wursthorn: Computational Aspects of the Isomorphism Problem R. Dipper, M. Geck, G. Hiss and G. Malle: Representations of Hecke algebras and finite groups of Lie type B. Eick and E. A. Orien: The groups of order 512 K. Lux and H. Pahlings: Computational aspects of representation theory of finite groups II G. O. Michler: High Performance Computations in Group Representation Theory G. Nebe: The structure of maximal finite primitive matrix groups W. Plesken: Presentations and representations of groups ' Chapter 2: Algorithmic Commutative Algebra and Algebraic Geometry p.177 E. Becker and J. Schmid: p.179 On the Real Nullstellensatz W. Decker, G.-M. Greuel and G. Pfister: p.193 Primary Decompositions: Algorithms and Comparisons A. Dolzmann, Th. Sturm and V. Weispfenning: p.227 Real Quantifier Elimination in Practice G. Kemper: p.255 Hilbert Series and Degree Bounds in Invariant Theory G. Kemper and G. Malle: p.271 Invariant rings and fields of finite groups B. Martin: p.289 Computing Versal Deformations with SINGULAR Th. Siebert: p.301 Algorithms for the computation of free resolutions Chapter 3: Algorithmic Group and Representation Theory p.319 F. M. Bleher, W. Kimmerle, K. W. Roggenkamp and M. Wursthorn: p.321 Computational Aspects of the Isomorphism Problem R. Dipper, M. Geck, G. Hiss and G. Malle: p.339 Representations of Hecke algebras and finite groups of Lie type B. Eick and E. A. Orien: p.387 The groups of order 512 K. Lux and H. Pahlings: p.389 Computational aspects of representation theory of finite groups II G. O. Michler: p.407 High Performance Computations in Group Representation Theory G. Nebe: p.425 The structure of maximal finite primitive matrix groups W. Plesken: p.431 Presentations and representations of groups '