cong.gd

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#W cong.gd                 The Congruence package                   Ann Dooms
#W                                                               Eric Jespers
#W                                                        Alexander Konovalov
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## InfoCongruence
##  
## We declare new Info class for algorithms from the Congruence package. It
## has 3 levels - 0, 1 and 2. Default level is 1, and it is used to display
## messages when the package is used to replace existing GAP methods.
## To change Info level to k, use command SetInfoLevel(InfoCongruence, k)
DeclareInfoClass("InfoCongruence");


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##
## IsCongruenceSubgroup( <G> )
## 
## We create category of congruence subgroups as a subcategory of matrix 
## groups, and declare properties that are used to distinguish several
## important classes of congruence subgroups
DeclareCategory( "IsCongruenceSubgroup", IsMatrixGroup );


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## IsPrincipalCongruenceSubgroup( <G> )
## 
## The principal congruence subgroup of level N consists of all matrices
## of the form   [ 1+N    N ]
##               [   N  1+N ]
##
DeclareProperty( "IsPrincipalCongruenceSubgroup", IsCongruenceSubgroup );


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## IsCongruenceSubgroupGamma0( <G> )
## 
## The congruence subgroup CongruenceSubgroupGamma0(N) consists of all matrices
## of the form   [   *    * ]
##               [   N    * ]
##
DeclareProperty( "IsCongruenceSubgroupGamma0", IsCongruenceSubgroup );


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## IsCongruenceSubgroupGammaUpper0( <G> )
## 
## The congruence subgroup CongruenceSubgroupGammaUpper0(N) consists of all matrices
## of the form   [   *    N ]
##               [   *    * ]
##
DeclareProperty( "IsCongruenceSubgroupGammaUpper0", IsCongruenceSubgroup );


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## IsCongruenceSubgroupGamma1( <G> )
## 
## The congruence subgroup CongruenceSubgroupGamma1(N) consists of all matrices
## of the form   [ 1+N    * ]
##               [   N  1+N ]
##
DeclareProperty( "IsCongruenceSubgroupGamma1", IsCongruenceSubgroup );


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## IsCongruenceSubgroupGammaUpper1( <G> )
## 
## The congruence subgroup CongruenceSubgroupGammaUpper1(N) consists of all matrices
## of the form   [ 1+N    N ]
##               [   *  1+N ]
##
DeclareProperty( "IsCongruenceSubgroupGammaUpper1", IsCongruenceSubgroup );


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## IsCongruenceSubgroupGammaMN( <G> )
## 
## The congruence subgroup CongruenceSubgroupGammaMN(M,N) consists of all matrices 
## of the form   [ 1+M    M ]
##               [   N  1+N ]
##
DeclareProperty( "IsCongruenceSubgroupGammaMN", IsCongruenceSubgroup );


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## IsIntersectionOfCongruenceSubgroups( <G> )
## 
## This property will be uses for subgroups of SL_2(Z) that were constructed
## as intersection of a finite number of congruence subgroups of types 
## CongruenceSubgroupGamma, CongruenceSubgroupGamma_0, 
## CongruenceSubgroupGamma^0, CongruenceSubgroupGamma_1,
## CongruenceSubgroupGamma^1 and CongruenceSubgroupGammaMN
##
DeclareProperty( "IsIntersectionOfCongruenceSubgroups", IsCongruenceSubgroup );


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## PrincipalCongruenceSubgroup( n )
## CongruenceSubgroupGamma0( n )
## CongruenceSubgroupGammaUpper0( n )
## CongruenceSubgroupGamma1( n )
## CongruenceSubgroupGammaUpper1( n )
## CongruenceSubgroupGammaMN( m, n )
##
## Declaration of global functions - constructors of congruence subgroups
##
DeclareOperation("PrincipalCongruenceSubgroup", [IsPosInt]);
DeclareOperation("CongruenceSubgroupGamma0", [IsPosInt]);
DeclareOperation("CongruenceSubgroupGammaUpper0", [IsPosInt]);
DeclareOperation("CongruenceSubgroupGamma1", [IsPosInt]);
DeclareOperation("CongruenceSubgroupGammaUpper1", [IsPosInt]);
DeclareOperation("CongruenceSubgroupGammaMN", [IsPosInt,IsPosInt]);


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## LevelOfCongruenceSubgroup( <G> )
##
## The (arithmetic) level of a congruence subgroup G is the smallest positive
## number N such that G contains the principal congruence subgroup of level N
##
DeclareAttribute( "LevelOfCongruenceSubgroup", IsCongruenceSubgroup );


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## LevelOfCongruenceSubgroupGammaMN( <G> )
##
## For the congruence subgroup GammaMN we need to store additionally
## two integers determining the 1st and 2nd lines of the matrix
##
DeclareAttribute( "LevelOfCongruenceSubgroupGammaMN", IsCongruenceSubgroup );


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## IndexInSL2Z( <G> )
##
## The index of a congruence subgroup in SL_2(Z) will be stored as its 
## attribute. This also will allow us to install a method for Index(G,H) when
## G is SL_2(Z) and H is a congruence subgroup. You should remember that we
## are working with the SL_2(Z), because it is available in GAP, and not with
## the PSL_2(Z) since the latter is not implemented in GAP.
##
DeclareAttribute( "IndexInSL2Z", IsCongruenceSubgroup );


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## IntersectionOfCongruenceSubgroups( <list of subgroups> )
##
## We declare special type of congruence subgroups that are intersections of
## a finite number congruence subgroups of types CongruenceSubgroupGamma, 
## CongruenceSubgroupGamma_0, CongruenceSubgroupGamma^0, CongruenceSubgroupGamma_1 
## and CongruenceSubgroupGamma^1. The list of subgroups defining this 
##  intersection will be stored in the attribute "DefiningCongruenceSubgroups".
##
DeclareGlobalFunction("IntersectionOfCongruenceSubgroups");
DeclareAttribute( "DefiningCongruenceSubgroups", 
                  IsCongruenceSubgroup );
                  
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# CanEasilyCompareCongruenceSubgroups( G, H )
#
DeclareOperation( "CanEasilyCompareCongruenceSubgroups", [ IsCongruenceSubgroup, IsCongruenceSubgroup ] );


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# CanReduceIntersectionOfCongruenceSubgroups( G, H )
#
# This function mimics the structure of the method for Intersection for
# congruence subgroups. It returns true, if their intersection can be reduced
# to one of the canonical congruence subgroups, and false otherwise, i.e. the
# intersection can be expressed only as IntersectionOfCongruenceSubgroups.
# This is used in IntersectionOfCongruenceSubgroups to reduce the list of
# canonical subgroups forming the intersection.
#
DeclareOperation( "CanReduceIntersectionOfCongruenceSubgroups", [ IsCongruenceSubgroup, IsCongruenceSubgroup ] );


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# NumeratorOfGFSElement( gfs, i )
#
# Returns the numerator of the i-th term of the generalised Farey sequence 
# gfs: for the 1st infinite entry returns -1, for the last one returns 1,
# for all other entries returns usual numerator.
#  
DeclareGlobalFunction( "NumeratorOfGFSElement" );


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# DenominatorOfGFSElement( gfs, i )
#
# Returns the denominator of the i-th term of the generalised Farey sequence 
# gfs: for both infinite entries returns 0, for the other ones returns usual 
# denominator.
# 
DeclareGlobalFunction( "DenominatorOfGFSElement" );


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# IsValidFareySymbol( fs )
#
# This function is used in FareySymbolByData to validate its output
# 
DeclareGlobalFunction( "IsValidFareySymbol" );


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# MatrixByEvenInterval( gfs, i )
#
DeclareGlobalFunction( "MatrixByEvenInterval" );


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# MatrixByOddInterval( gfs, i ) 
#
DeclareGlobalFunction( "MatrixByOddInterval" );


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# MatrixByFreePairOfIntervals( gfs, k, kp )
#
DeclareGlobalFunction( "MatrixByFreePairOfIntervals" );


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#E
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