#18548: Fix a bug introduced in #17792
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Reporter: mmasdeu | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.8
Component: modular forms | Keywords:
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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Ticket #17792 implemented the solution for the word problem for congruence
subgroups of (P)SL2(Z). However, the element represented by the output
word sometimes differs from the original element by at most one generator.
Here is an example:
{{{
sage: G = Gamma0(10)
sage: F = G.farey_symbol()
sage: g = G([-701,-137,4600,899])
sage: word = F.word_problem(g, output = 'syllables')
sage: g1 = prod(F.generators()[i]**a for i,a in word)
sage: g == g1 or g * G([-1,0,0,-1]) == g1
False # Should be True
}}}
This ticket solves this. I have ran the code below to test:
{{{
for N in range(2,500):
G = Gamma0(N)
print "N = %s"%N
gens = G.generators()
F = G.farey_symbol()
i = 0
I = G([1,0,0,1])
E = G([-1,0,0,-1])
while i < 200:
c = ZZ.random_element(1000)
d = ZZ.random_element(1000)
if gcd(c*N,d) > 1:
continue
i += 1
_,a,b = xgcd(d,-c * N)
g = G([a,b,c*N,d])
wd = F.word_problem(g, output = 'syllables')
g1 = prod(gens[j]**n for j,n in wd)
assert g * g1**-1 == I or g * g1**-1 == E
}}}
(and also a modification to check Gamma_1(N).
--
Ticket URL: <http://trac.sagemath.org/ticket/18548>
Sage <http://www.sagemath.org>
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