chap0.txt

  
  
                                   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:[email protected]
      Homepage: http://homepages.vub.ac.be/~andooms
      Address:  Department of Mathematics, Vrije Universiteit Brussel
                Pleinlaan 2, Brussels, B-1050 Belgium
  
  
  Eric Jespers
      Email:    mailto:[email protected]
      Homepage: http://homepages.vub.ac.be/~efjesper
      Address:  Department of Mathematics, Vrije Universiteit Brussel
                Pleinlaan 2, Brussels, B-1050 Belgium
  
  
  Alexander Konovalov
      Email:    mailto:[email protected]
      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:[email protected]
      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