Bosma / Cannon Discovering Mathematics with Magma

Reducing the Abstract to the Concrete
1. Auflage 2007
ISBN: 978-3-540-37634-7
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 19, 364 Seiten

Reihe: Algorithms and Computation in Mathematics

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.

Autoren/Hrsg.

Bosma, Wieb

Cannon, John

Inhaltsverzeichnis

1;Preface;6
2;Magma: The project;8
3;Discovering mathematics: About this volume;14
4;How to read the Magma code;20
5;Contents;24
6;Some computational experiments in number theory;26
6.1;1 Introduction;26
6.2;2 Covering systems;27
6.3;3 Covering systems and explicit primality tests;32
6.4;4 The totient function;40
6.5;5 Class number relations;46
6.6;References;53
7;Applications of the class field theory of global fields;56
7.1;1 Introduction;56
7.2;2 Number fields;57
7.3;3 Global function fields;70
7.4;4 Applications;77
7.5;References;86
8;Some ternary Diophantine equations of signature (n, n, 2);88
8.1;1 Introduction;88
8.2;2 Proof of Proposition 1.3;90
8.3;3 Construction of parametrising curves;91
8.4;4 The equation x5 + y5 = Dz2;93
8.5;5 Deciding local solvability;97
8.6;6 Mordell–Weil groups of elliptic curves;104
8.7;7 Chabauty methods using elliptic curves;109
8.8;8 The equations xn + yn = Dz2 for n = 6, 7, 9, 11, 13, 17;113
8.9;References;115
9;Studying the Birch and Swinnerton-Dyer conjecture for modular abelian varieties using Magma;118
9.1;1 Introduction;118
9.2;2 Modular abelian varieties;119
9.3;3 The Birch and Swinnerton-Dyer Conjecture;122
9.4;4 Some computational results;125
9.5;5 Modular symbols;126
9.6;6 Visibility theory;129
9.7;7 Computing special values of modular L-function;130
9.8;8 Computing Tamagawa numbers;131
9.9;9 Computing the torsion subgroup;132
9.10;10 A divisor and multiple of the order of the Shafarevich – Tate group;132
9.11;11 An element of the Shafarevich–Tate group that becomes visible at higher level;133
9.12;12 Complete Magma log;136
9.13;References;139
10;Computing with the analytic Jacobian of a genus 2 curve;142
10.1;1 Introduction;142
10.2;2 Finding genus 2 CM curves defined over the rationals;145
10.3;3 Isogenies;152
10.4;References;159
11;Graded rings and special K3 surfaces;162
11.1;1 Introduction;162
11.2;2 Elementary example;165
11.3;3 Graded rings of polarised varieties;168
11.4;4 Subcanonical curves;169
11.5;5 K3 database;171
11.6;6 Simple degenerations of the famous 95;173
11.7;7 Unprojection;178
11.8;8 Special K3 surfaces in Fletcher’s 84;180
11.9;References;183
12;Constructing the split octonions;186
12.1;1 Introduction;186
12.2;2 Structure constant algebras;188
12.3;3 Lie algebras of type D4 and E6;191
12.4;4 Triality;194
12.5;5 The Lie algebra of type G2;196
12.6;6 The split octonions;198
12.7;7 The quadratic form;202
12.8;8 The Chevalley groups of type G2;205
12.9;References;210
13;Support varieties for modules;212
13.1;1 Introduction;212
13.2;2 Notes on projectivity;214
13.3;3 Support varieties and rank varieties;215
13.4;4 Finding points on the variety;217
13.5;5 Computing the variety from a set of points;224
13.6;6 Varieties of truncated syzygy modules;226
13.7;References;228
14;When is projectivity detected on subalgebras?;230
14.1;1 Introduction;230
14.2;2 Criterion for projectivity;231
14.3;3 Basic algebras and homological algebra on the computer;233
14.4;4 Support varieties for modules over group algebras;234
14.5;5 Some notes on cohomology and computations;236
14.6;6 An algebra whose projective modules are detected on proper subalgebras;238
14.7;7 An example in which projectivity is not detected on subalgebras;241
14.8;References;244
15;Cohomology and group extensions in Magma;246
15.1;1 Introduction;246
15.2;2 Computing cohomology groups;247
15.3;3 Finding group extensions;261
15.4;References;266
16;Computing the primitive permutation groups of degree less than 1000;268
16.1;1 Some background;268
16.2;2 Mathematical preliminaries;269
16.3;3 Determining conjugacy;271
16.4;4 Maximal irreducible subgroups of GL(4, 5);280
16.5;5 The main algorithm;282
16.6;6 Results;283
16.7;References;284
17;Computer aided discovery of a fast algorithm for testing conjugacy in braid groups;286
17.1;1 Introduction;286
17.2;2 Background: braid groups and testing conjugacy;287
17.3;3 Coming across another class invariant;293
17.4;4 On the way to a proof;299
17.5;5 Computing minimal simple elements;306
17.6;6 An application: key recovery;307
17.7;References;310
18;Searching for linear codes with large minimum distance;312
18.1;1 Introduction;312
18.2;2 Computing the minimum weight;314
18.3;3 Constructing new codes from old ones;326
18.4;4 Searching for good codes;328
18.5;5 Conclusion;337
18.6;6 Acknowledgements;337
18.7;References;338
19;Colouring planar graphs;340
19.1;1 Introduction;340
19.2;2 k-Flows in graphs;341
19.3;3 Planar graphs;342
19.4;4 k-Flows and k-colouring in planar graphs;347
19.5;5 Finding nowhere-zero k-Flows;348
19.6;6 Testing for nowhere-zero k-Flows;350
19.7;7 Conclusion;354
19.8;References;354
20;Appendix: The Magma language;356
20.1;Introduction;356
20.2;1 Basics;356
20.3;2 Sets and sequences;364
20.4;3 Tuples;372
20.5;4 Creating functions and procedures;373
20.6;5 Loops: for, while, and repeat;374
20.7;6 Maps;377
20.8;References;381
21;Index;382