Congruence Congruence subgroups of SL_2(ℤ) Version 1.2.1 26 September 2017 Ann Dooms Eric Jespers Alexander Konovalov Helena Verrill Ann Dooms Email: mailto:andooms@vub.ac.be Homepage: http://homepages.vub.ac.be/~andooms Address: Department of Mathematics, Vrije Universiteit Brussel Pleinlaan 2, Brussels, B-1050 Belgium Eric Jespers Email: mailto:efjesper@vub.ac.be Homepage: http://homepages.vub.ac.be/~efjesper Address: Department of Mathematics, Vrije Universiteit Brussel Pleinlaan 2, Brussels, B-1050 Belgium Alexander Konovalov Email: mailto:alexander.konovalov@st-andrews.ac.uk Homepage: https://alexk.host.cs.st-andrews.ac.uk Address: School of Computer Science University of St Andrews Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland Helena Verrill Email: mailto:verrill@math.lsu.edu Homepage: http://www.math.lsu.edu/~verrill/ Address: Department of Mathematics Louisiana State University Baton Rouge, Louisiana, 70803-4918 USA ------------------------------------------------------- Abstract The GAP package Congruence provides functionality to work with congruence subgroups of SL_2(ℤ). ------------------------------------------------------- Copyright © 2006-2017 by Ann Dooms, Eric Jespers, Alexander Konovalov and Helena Verrill. Congruence is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site http://www.gnu.org/licenses/gpl.html. If you obtained Congruence, we would be grateful for a short notification sent to one of the authors. If you publish a result which was partially obtained with the usage of Congruence, please cite it in the following form: A. Dooms, E. Jespers, A. Konovalov and H. Verrill. Congruence --- Congruence subgroups of SL_2(ℤ), Version 1.2.1; 2017 (http://www.cs.st-andrews.ac.uk/~alexk/congruence/). ------------------------------------------------------- Acknowledgements We are very grateful to Mong-Lung Lang, Chong-Hai Lim and Ser Peow Tan for their comments provided while implementing algorithms from [LLT95a] and [LLT95b], and to Francqui Stichting (Belgium) for the support of the third author. ------------------------------------------------------- Contents (Congruence) 1 Introduction 1.1 General aims of Congruence package 1.2 Installation and system requirements 2 Construction of congruence subgroups 2.1 Construction of congruence subgroups 2.1-1 PrincipalCongruenceSubgroup 2.1-2 CongruenceSubgroupGamma0 2.1-3 CongruenceSubgroupGammaUpper0 2.1-4 CongruenceSubgroupGamma1 2.1-5 CongruenceSubgroupGammaUpper1 2.1-6 IntersectionOfCongruenceSubgroups 2.2 Properties of congruence subgroups 2.2-1 IsPrincipalCongruenceSubgroup 2.2-2 IsCongruenceSubgroupGamma0 2.2-3 IsCongruenceSubgroupGammaUpper0 2.2-4 IsCongruenceSubgroupGamma1 2.2-5 IsCongruenceSubgroupGammaUpper1 2.2-6 IsIntersectionOfCongruenceSubgroups 2.3 Attributes of congruence subgroups 2.3-1 LevelOfCongruenceSubgroup 2.3-2 IndexInSL2Z 2.3-3 DefiningCongruenceSubgroups 2.4 Operations for congruence subgroups 2.4-1 Random 2.4-2 \in 2.4-3 CanEasilyCompareCongruenceSubgroups 2.4-4 IsSubset 2.4-5 Index 3 Farey symbols and their properties 3.1 Construction of Farey symbols 3.1-1 FareySymbolByData 3.1-2 IsValidFareySymbol 3.2 Properties of Farey symbols 3.2-1 GeneralizedFareySequence 3.2-2 NumeratorOfGFSElement 3.2-3 DenominatorOfGFSElement 3.2-4 LabelsOfFareySymbol 4 Farey symbols for congruence subgroups 4.1 Computation of the Farey symbol for a finite index subgroup 4.1-1 FareySymbol 4.2 Computation of generators of a finite index subgroup from its Farey symbol 4.2-1 MatrixByEvenInterval 4.2-2 MatrixByOddInterval 4.2-3 MatrixByFreePairOfIntervals 4.2-4 GeneratorsByFareySymbol 4.2-5 GeneratorsOfGroup 4.3 Other properties derived from Farey symbols 4.3-1 IndexInPSL2ZByFareySymbol 5 Service functions of the Congruence package 5.1 Additional information displayed by Congruence algorithms 5.1-1 InfoCongruence