I wrote the following function, which does the job. Function below
takes as input a positive integer N and outputs two objects: the first
output is a list [g_i] of hyperbolic elements in Gamma0(N) which
generate the abelianized (Gamma0(N)_hyp)_ab of the quotient
Gamma0(N)_hyp of Gamma0(N) by the subgroup generated by parabolic and
elliptic elements. Group (Gamma0(N)_hyp)_ab is naturally isomorphic to
H_1(X_0(N), ZZ) and is therefore free of rank 2g over ZZ, where g
stands for the genus of the curve. It can also be regarded as the
cuspidal subspace of the module of modular symbols of level N and
weight 2. The second output is the matrix of the coordinates of each
of the modular symbols {oo, g_i(oo)}, expressed in the basis given by
Sage of the space of modular symbols of level N and weight 2.
Since I don't have much experience in programming, the algorithm
below is most probably not the most efficient one, I'll be happy if
anybody suggests an improved version of it. In any case, I computed
the elements g_i in such a way that the entry c in the lower-left
corner is N or -N. This is something I needed in order to make my
subsequent computations faster.
def hypgens(N):
a=1;
G=Gamma0(N);
M=ModularSymbols(G);
d=M.cuspidal_subspace().dimension();
Gens=[];
GensM=[];
Mat=[];
MatAux=[];
rnext=matrix(Mat).rank();
while rnext<d:
a=a+1;
rant=rnext;
g=G.are_equivalent_cusps(Cusp(Infinity),Cusp(a/N),trans=True);
if g:
if (g.a()+g.d()).abs()>2:
m=M.modular_symbol([Infinity,g.a()/g.c()]);
c=m.list();
MatAux.append(c);
rnext=matrix(MatAux).rank();
if rnext>rant:
if rnext <= d:
Mat.append(c);
Gens.append(g);
GensM.append(m);
return Gens,Matrix(Mat)
-------------
Example:
sage: Gens11,Mat11=hypgens(11); Gens11;Mat11
[[ 2 -1][11 -5], [ 3 1][11 4]]
[ 0 0 1]
[ 0 -1 1]
Given a cuspidal modular symbol m in ModularSymbols(11), if one now
wishes to find an element g in Gamma0(11) such that {oo, g(oo)} = m,
one can do the following:
sage: M=ModularSymbols(11);
m = M.modular_symbol([8/177,-61/571]);
X = Mat11.transpose().solve_right(vector(m.list()));X
g=prod(Gens11[i]^(X[i]) for i in [0..len(X)-1]);g
[-1453 377]
[-3091 802]
And one also has the factoization of this matrix as a product of the
generating matrices g_i with c_i= 11 or -11.
----
On 21 dic, 18:37, victor <[email protected]> wrote:
> Let m be a modular symbol for the congruence subgroup G=Gamma0(N) for
> some N.
>
> If one assumes m is cuspidal, there exist elements g in G such that m
> is equivalent to the symbol {0,g(0)}.
>
> How can I compute one such g with sage? If possible, I'd like to find
> g with as small coefficients as possible. I'd particularly interested
> in finding g=[[a,b],[c,d]] with c small.
>
> If m={c1,c2} then we can just type
> G.are_equivalent_cusps(c1,c2,trans=True) and the answer is True,
> because cuspidality means that c1 and c2 are equivalent cusps under G.
>
> How can we compute g when, say, m={c1,c2}+{c3,c4} is cuspidal but c1
> is not equivalent to c2, and c3 is not equivalent to c4? Is there an
> optimal way of computing it?
--
To post to this group, send email to [email protected]
To unsubscribe from this group, send email to
[email protected]
For more options, visit this group at
http://groups.google.com/group/sage-support
URL: http://www.sagemath.org
