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#############################################################################
##
#W random.gi The Congruence package Ann Dooms
#W Eric Jespers
#W Olexandr Konovalov
##
##
#############################################################################
##
## This file contains implementations of methods to construct random elements
## of congruence subgroups CongruenceSubgroupGamma, CongruenceSubgroupGamma0,
## CongruenceSubgroupGammaUpper0, CongruenceSubgroupGamma1 and
## CongruenceSubgroupGammaUpper1.
## The idea is to select two random entries a and b in the same row or column
## of the matrix, such that a and b will satisfy the requirements arising
## from the congruence subgroup. For example, for the principal congruence
## subgroup we will select a and b as follows:
## a := 1 + n * Random( [ -10 .. 10 ] );
## b := n * Random( [ -10 .. 10 ] );
## After this we can find such x and y for the other row (or column) of the
## matrix that its determinant will be equal to one. If the resulting matrix
## will be not in the congruence subgroup because of not suitable x and y,
## we will repeat this process for another a and b until we will find
## suitable x and y.
## For each type of congruence subgroups, we provide one- and two-argument
## versions of Random. The one-argument version uses Random( [ -10 .. 10 ] )
## to generate a and b, ## and in the two-argument version Random([ -m..m ])
## will be used, where m is given by the second argument.
#############################################################################
##
## The principal congruence subgroup of level N consists of all matrices
## of the form [ 1+N N ]
## [ N 1+N ]
##
InstallMethod( Random,
"for a principal congruence subgroup",
[ IsPrincipalCongruenceSubgroup ],
0,
function( G )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -10 .. 10 ] );
b := n * Random( [ -10 .. 10 ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and
IsInt( -gcd.coeff2/n ) and
IsInt( (gcd.coeff1-1)/n );
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
InstallOtherMethod( Random,
"for a principal congruence subgroup",
[ IsPrincipalCongruenceSubgroup, IsPosInt ],
0,
function( G, m )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -m .. m ] );
b := n * Random( [ -m .. m ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and
IsInt( -gcd.coeff2/n ) and
IsInt( (gcd.coeff1-1)/n );
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
#############################################################################
##
## The congruence subgroup CongruenceSubgroupGamma0(N) consists of all matrices
## of the form [ * * ]
## [ N * ]
##
InstallMethod( Random,
"for a congruence subgroup CongruenceSubgroupGamma0",
[ IsCongruenceSubgroupGamma0 ],
0,
function( G )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := Random( [ -n*10 .. n*10 ] );
b := n * Random( [ -10 .. 10 ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1;
return [ [ a, -gcd.coeff2 ],
[ b, gcd.coeff1 ] ];
end);
InstallOtherMethod( Random,
"for a congruence subgroup CongruenceSubgroupGamma0",
[ IsCongruenceSubgroupGamma0, IsPosInt ],
0,
function( G, m )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := Random( [ -n*m .. n*m ] );
b := n * Random( [ -m .. m ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1;
return [ [ a, -gcd.coeff2 ],
[ b, gcd.coeff1 ] ];
end);
#############################################################################
##
## The congruence subgroup CongruenceSubgroupGammaUpper0(N) consists of all matrices
## of the form [ * N ]
## [ * * ]
##
InstallMethod( Random,
"for a congruence subgroup CongruenceSubgroupGammaUpper0",
[ IsCongruenceSubgroupGammaUpper0 ],
0,
function( G )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := Random( [ -n*10 .. n*10 ] );
b := n * Random( [ -10 .. 10 ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1;
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
InstallOtherMethod( Random,
"for a congruence subgroup CongruenceSubgroupGammaUpper0",
[ IsCongruenceSubgroupGammaUpper0, IsPosInt ],
0,
function( G, m )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := Random( [ -n*m .. n*m ] );
b := n * Random( [ -m .. m ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1;
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
#############################################################################
##
## The congruence subgroup CongruenceSubgroupGamma1(N) consists of all matrices
## of the form [ 1+N * ]
## [ N 1+N ]
##
InstallMethod( Random,
"for a congruence subgroup CongruenceSubgroupGamma1",
[ IsCongruenceSubgroupGamma1 ],
0,
function( G )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -10 .. 10 ] );
b := n * Random( [ -10 .. 10 ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and IsInt( (gcd.coeff1-1)/n );
return [ [ a, -gcd.coeff2 ],
[ b, gcd.coeff1 ] ];
end);
InstallOtherMethod( Random,
"for a congruence subgroup CongruenceSubgroupGamma1",
[ IsCongruenceSubgroupGamma1, IsPosInt ],
0,
function( G, m )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -m .. m ] );
b := n * Random( [ -m .. m ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and IsInt( (gcd.coeff1-1)/n );
return [ [ a, -gcd.coeff2 ],
[ b, gcd.coeff1 ] ];
end);
#############################################################################
##
## The congruence subgroup CongruenceSubgroupGammaUpper1(N) consists of all matrices
## of the form [ 1+N N ]
## [ * 1+N ]
##
InstallMethod( Random,
"for a congruence subgroup CongruenceSubgroupGammaUpper1",
[ IsCongruenceSubgroupGammaUpper1 ],
0,
function( G )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -10 .. 10 ] );
b := n * Random( [ -10 .. 10 ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and IsInt( (gcd.coeff1-1)/n );
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
InstallOtherMethod( Random,
"for a congruence subgroup CongruenceSubgroupGammaUpper1",
[ IsCongruenceSubgroupGammaUpper1, IsPosInt ],
0,
function( G, m )
local n, a, b, gcd;
n := LevelOfCongruenceSubgroup( G );
repeat
a := 1 + n * Random( [ -m .. m ] );
b := n * Random( [ -m .. m ] );
gcd := Gcdex( a, b );
until gcd.gcd = 1 and IsInt( (gcd.coeff1-1)/n );
return [ [ a, b ],
[ -gcd.coeff2, gcd.coeff1 ] ];
end);
#############################################################################
##
#E
##
|
