Discovering Mathematics with Magma: Reducing the Abstract to the Concrete (Algorithms and Computation in Mathematics Book 19)

This volume celebrates the first decade of the Computer Algebra system Magma. With a design based on the ontology and semantics of algebra, Magma enables users to rapidly formulate and perform calculations in the more abstract parts of mathematics. This book introduces the reader to the role Magma plays in advanced mathematical research through 14 case studies which, in most cases, describe computations underpinning new theoretical results. The authors of the chapters were chosen both for their expertise in the particular field and for their innovative use of Magma. Although by no means exhaustive, the topics range over much of Magma's coverage of algorithmic algebra: from number theory and algebraic geometry, via representation theory and group theory to some branches of discrete mathematics and graph theory. A basic introduction to the Magma language is given in an appendix. The book is simultaneously an invitation to learn a new programming language in the context of contemporary research problems, and an exposition of the types of problem that can be investigated using computational algebra. The appearance of this volume celebrates the?rst decade of Magma, a new computeralgebrasystemlaunchedattheFirstMagmaConferenceonCom- tational Algebra held at Queen Mary and West?eld College, London, August 1993. This book introduces the reader to the role Magma plays in advanced mathematical research. Each paper examines how the computer can be used to gain insight into either a single problem or a small group of closely related problems. The intention is to present su?cient detail so that a reader can (a), gain insight into the mathematical questions that are the origin of the problems,and(b),developanunderstandingastohowsuchcomputations are speci?edinMagma.Itishopedthatthereaderwillcometoarealisationofthe important rolethatcomputational algebracanplayinmathematical research. Readers not primarily interested in using Magma will easily acquire the skills needed to undertake basic programming in Magma, while experienced Magma users can learn both mathematics and advanced computational methods in areas related to their own. The core of the volume comprises 14 papers. The authors were invited to submit articles on designated topics and these articles were then reviewed by referees. Although by no means exhaustive, the topics range over a consid- ablepartofMagma'scoverageofalgorithmicalgebra:fromnumbertheoryand algebraicgeometry,viarepresentationtheoryandcomputationalgrouptheory to some branches of discrete mathematics and graph theory. The papers are preceded by an outline of the Magma project, a brief summary of the papers and some instructions on reading the Magma code. A basic introduction to the Magma language is given in an appendix. Theeditorsexpresstheirgratitudetothecontributorstothisvolume,both for the work put into producing the papers and for theirpatience. Front Matter....Pages I-XXIV Some computational experiments in number theory....Pages 1-30 Applications of the class field theory of global fields....Pages 31-62 Some ternary Diophantine equations of signature ( n , n , 2)....Pages 63-91 Studying the Birch and Swinnerton-Dyer conjecture for modular abelian varieties using Magma....Pages 93-116 Computing with the analytic Jacobian of a genus 2 curve....Pages 117-135 Graded rings and special K3 surfaces....Pages 137-159 Constructing the split octonions....Pages 161-185 Support varieties for modules....Pages 187-204 When is projectivity detected on subalgebras?....Pages 205-220 Cohomology and group extensions in Magma....Pages 221-241 Computing the primitive permutation groups of degree less than 1000....Pages 243-260 Computer aided discovery of a fast algorithm for testing conjugacy in braid groups....Pages 261-285 Searching for linear codes with large minimum distance....Pages 287-313 Colouring planar graphs....Pages 315-330 Appendix: The Magma language....Pages 331-356 Back Matter....Pages 356-374 Based on the ontology and semantics of algebra, the computer algebra system Magma enables users to rapidly formulate and perform calculations in abstract parts of mathematics. Edited by the principal designers of the program, this book explores Magma. Coverage ranges from number theory and algebraic geometry, through representation theory and group theory to discrete mathematics and graph theory. Includes case studies describing computations underpinning new theoretical results.