Algorithmic Algebra and Number Theory

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.

The book contains contributions by many top class researchers in algorithmic number theory.

Klappentext
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.

Inhalt
Algorithmic Algebraic Number Theory.- Sieving Methods for Class Group Computation.- Arithmetic of Modular Curves and Applications.- Local and Global Ramification Properties of Elliptic Curves in Characteristics Two and Three.- Techniques for the Computation of Galois Groups.- Fortschritte in der inversen Galoistheorie.- From Class Groups to Class Fields.- A Gross-Zagier Formula for Function Fields.- Extremal Lattices.- Algorithmic Commutative Algebra and Algebraic Geometry.- On the Real Nullstellensatz.- Primary Decomposition: Algorithms and Comparisons.- Real Quantifier Elimination in Practice.- Hilbert Series and Degree Bounds in Invariant Theory.- Invariant Rings and Fields of Finite Groups.- Computing Versal Deformations with Singular.- Algorithms for the Computation of Free Resolutions.- Algorithmic Group and Representation Theory.- Computational Aspects of the Isomorphism Problem.- Representations of Heeke Algebras and Finite Groups of Lie Type.- The Groups of Order 512.- Computational Aspects of Representation Theory of Finite Groups II.- High Performance Computations in Group Representation Theory.- The Structure of Maximal Finite Primitive Matrix Groups.- Presentations and Representations of Groups.

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