[Next][Prev] [Right] [Left] [Up] [Index] [Root]
Subsections
If G is a subgroup of finite index in PSL2(Z), then
returns a sequence of coset representatives of G in PSL2(Z).
Returns a sequence of generators of the congruence subgroup G.
For a congruence subgroup G, and an element g of G,
this function returns a sequence of integers corresponding to an
expression for g in terms of a fixed set of generators for G.
Let L be the list of generators for G output by the function
Generators. Then the return sequence
[e1n1, e2n2, ..., em nm],
where ni are positive integers, and ei=1 or -1, means that
g=L[n1]e1L[n2]e2 ... L[nm]em.
Note that since the computation is in PSL2(R), this equality
only holds up to multiplication by +- 1.
The genus of the upper half plane quotiented by the congruence
subgroup G.
For G a subgroup of PSL2(Z)
returns a sequence of points in the Upper Half plane which are the
vertices of a fundamental domain for G.
In this example we compute a set of generators for
Γ0(12).
> G := CongruenceSubgroup(0,12);
> Generators(G);
[
[1 1]
[0 1],
[ 5 -1]
[36 -7],
[ 5 -4]
[ 24 -19],
[ 7 -5]
[ 24 -17],
[ 5 -3]
[12 -7]
]
> C := CosetRepresentatives(G);
> H<i,r> := UpperHalfPlaneWithCusps();
> triangle := [H|Infinity(),r,r-1];
> translates := [g*triangle : g in C];
This example illustrates how any element of a congruence
subgroup can be written in terms of the set of generators
output by the generators function.
> N := 34;
> characters := DirichletGroup(N,CyclotomicField(EulerPhi(N)));
> char := Elements(characters)[5];
> G := CongruenceSubgroup([N,Conductor(char),1],char);
>
> // We can create a list of generators:
> gens := Generators(G);
> #gens;
21
> // given an element of G,
> g := G![ 21, 4, 68, 13 ];;
> // we can find g in terms of these generators:
> FindWord(G,g);
[ -8, 1 ]
> // This means that up to sign, g = gens[8]^(-1)*gens[1],
> // which can be verified as follows:
> gens[8]^(-1)*gens[1];
[-21 -4]
[-68 -13]
Returns a sequence of inequivalent cusps of
the congruence subgroup G.
Returns the width of x as a cusp of the congruence
subgroup G.
EllipticPoints(G,H) : GrpPSL2, SpcHyp -> [SpcHypElt]
Returns a list of inequivalent elliptic points for the congruence subgroup G.
A second argument may be given to specify the upper half plane H
containing these elliptic points.
We can compute a set of representative cusps for
Γ1(12), and their widths as follows:
> G := CongruenceSubgroup(0,12);
> Cusps(G);
[
oo,
0,
1/6,
1/4,
1/3,
1/2
]
> Widths(G);
[ 1, 12, 1, 3, 4, 3 ]
> // Note that the sum of the cusp widths is the same as the Index:
> &+Widths(G);
24
> Index(G);
24
In the following example we find which group Γ0(N) has the
most elliptic points for N less than 20, and list the elliptic points
in this case.
> H := UpperHalfPlaneWithCusps();
> [#EllipticPoints(Gamma0(N),H) : N in [1..20]];
[ 2, 1, 1, 0, 2, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0 ]
> // find the index where the maximal number of elliptic points is attained:
> Max($1);
4 13
> // find the elliptic points for Gamma0(13):
> EllipticPoints(Gamma0(13));
[
5/13 + (1/13)*root(-1),
8/13 + (1/13)*root(-1),
7/26 + (1/26)*root(-3),
19/26 + (1/26)*root(-3)
]
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
