Algorithmic Algebra and Number Theory : Selected Papers From a Conference Held at the University of Heidelberg in October 1997

This book contains 22 lectures presented at the final conference of the German research program "Algorithmic Number Theory and Algebra 1991-1997", sponsored by the Deutsche Forschungsgemeinschaft. The purpose of this research program and the meeting was to bring together developers of computer algebra software and researchers using computational methods to gain insight into experimental problems and theoretical questions in algebra and number theory. The book gives an overview on algorithmic methods and results obtained during this period mainly in algebraic number theory, commutative algebra and algebraic geometry, and group and representation theory. Some of the articles illustrate the current state of the computer algebra systems developed with support from the research program, for example KANT and LiDIA for algebraic number theory, SINGULAR, REDLOG and INVAR for commutative algebra and invariant theory respectively, and GAP, SYSYPHOS and CHEVIE for group and representation theory. Front Matter....Pages I-VIII Front Matter....Pages 1-1 Sieving Methods for Class Group Computation....Pages 3-10 Arithmetic of Modular Curves and Applications....Pages 11-48 Local and Global Ramification Properties of Elliptic Curves....Pages 49-64 Techniques for the Computation of Galois Groups....Pages 65-77 Fortschritte in der inversen Galoistheorie....Pages 79-102 From Class Groups to Class Fields....Pages 103-119 A Gross-Zagier Formula for Function Fields....Pages 121-137 Extremal Lattices....Pages 139-170 Front Matter....Pages 171-171 On the Real Nullstellensatz....Pages 173-185 Primary Decomposition: Algorithms and Comparisons....Pages 187-220 Real Quantifier Elimination in Practice....Pages 221-247 Hilbert Series and Degree Bounds in Invariant Theory....Pages 249-263 Invariant Rings and Fields of Finite Groups....Pages 265-281 Computing Versal Deformations with Singular....Pages 283-293 Algorithms for the Computation of Free Resolutions....Pages 295-310 Front Matter....Pages 311-311 Computational Aspects of the Isomorphism Problem....Pages 313-329 Representations of Heeke Algebras and Finite Groups of Lie Type....Pages 331-378 The Groups of Order 512....Pages 379-380 Computational Aspects of Representation Theory of Finite Groups II....Pages 381-397 High Performance Computations in Group Representation Theory....Pages 399-415 Front Matter....Pages 311-311 The Structure of Maximal Finite Primitive Matrix Groups....Pages 417-422 Presentations and Representations of Groups....Pages 423-434 Back Matter....Pages 435-437 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. This text gives an overview on algorithmic methods and results obtained during the period 1991-1997 mainly in algebraic geometry, and group and representation theory.