#10510: ModularSymbols(Gamma1(29),
2).cuspidal_subspace().hecke_algebra().basis()
doesn't terminate
-----------------------------+----------------------------------------------
Reporter: mderickx | Owner: craigcitro
Type: defect | Status: new
Priority: major | Milestone:
Component: modular forms | Keywords:
Author: mderickx | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
-----------------------------+----------------------------------------------
Description changed by was:
Old description:
> It also doesn't terminate for other primes then 29, but here is a more
> detailed example of the case 29 which should illustrate everything
>
> The sourcecode which determines the basis is currently:
>
> {{{
> def basis(self):
> try:
> return self.__basis_cache
> except AttributeError:
> pass
> level = self.level()
> bound = self.__M.hecke_bound()
> dim = self.__M.rank()
> if dim == 0:
> basis = []
> elif dim == 1:
> basis = [self.hecke_operator(1)]
> else:
> span = [self.hecke_operator(n) for n in range(1, bound+1) if not
> self.is_anemic() or gcd(n, level) == 1]
> rand_max = 5
> while True:
> # Project the full Hecke module to a random submodule to ease
> the HNF reduction.
> v = (ZZ**dim).random_element(x=rand_max)
> proj_span = matrix([T.matrix()*v for T in
> span])._clear_denom()[0]
> proj_basis = proj_span.hermite_form()
> if proj_basis[dim-1] == 0:
> # We got unlucky, choose another projection.
> rand_max *= 2
> continue
> # Lift the projected basis to a basis in the Hecke algebra.
> trans = proj_span.solve_left(proj_basis)
> basis = [sum(c*T for c,T in zip(row,span) if c != 0) for row
> in trans[:dim]]
> break
>
> self.__basis_cache = tuple(basis)
> return basis
> }}}
>
> Now dim equals 44 and the hecke bound 140 in this example:
> {{{
> sage: G29=Gamma1(29)
> sage: M=ModularSymbols(G29,2)
> sage: m=M.cuspidal_subspace()
> sage: m.rank()
> 44
> sage: m.hecke_bound()
> 140
> }}}
>
> Now we see that the hecke algebra has rank 22 over ZZ
> {{{
> sage: HA=m.hecke_algebra()
> sage: span = [HA.hecke_operator(n) for n in range(1, 140+1) if not
> HA.is_anemic() or gcd(n, 29) == 1]
> sage: (ZZ^(44^2)).span([i.matrix().list() for i in span])
> Free module of degree 1936 and rank 22 over Integer Ring
> Echelon basis matrix:
> 22 x 1936 dense matrix over Rational Field
> }}}
> So as the comment say's we are unlucky since proj_basis[43] will always
> be zero since the hecke algebra has only rank 22
> {{{
> if proj_basis[dim-1] == 0:
> # We got unlucky, choose another projection.
> rand_max *= 2
> continue
> }}}
> I suspect there is a problem in that the rank of m is 44 over QQ. But
> that the rank of the hecke algebra will be the dimension of m as a
> complex vector space so dim should actually be half of what it is now.
>
> I don't know enough of modular symbols yet to know how it should be done
> in other weight's and other modular groups but I suspect there will be
> problems there also.
New description:
It also doesn't terminate for other primes then 29, but here is a more
detailed example of the case 29 which should illustrate everything
The sourcecode which determines the basis is currently:
{{{
def basis(self):
try:
return self.__basis_cache
except AttributeError:
pass
level = self.level()
bound = self.__M.hecke_bound()
dim = self.__M.rank()
if dim == 0:
basis = []
elif dim == 1:
basis = [self.hecke_operator(1)]
else:
span = [self.hecke_operator(n) for n in range(1, bound+1) if not
self.is_anemic() or gcd(n, level) == 1]
rand_max = 5
while True:
# Project the full Hecke module to a random submodule to ease
the HNF reduction.
v = (ZZ**dim).random_element(x=rand_max)
proj_span = matrix([T.matrix()*v for T in
span])._clear_denom()[0]
proj_basis = proj_span.hermite_form()
if proj_basis[dim-1] == 0:
# We got unlucky, choose another projection.
rand_max *= 2
continue
# Lift the projected basis to a basis in the Hecke algebra.
trans = proj_span.solve_left(proj_basis)
basis = [sum(c*T for c,T in zip(row,span) if c != 0) for row
in trans[:dim]]
break
self.__basis_cache = tuple(basis)
return basis
}}}
Now dim equals 44 and the hecke bound 140 in this example:
{{{
sage: G29=Gamma1(29)
sage: M=ModularSymbols(G29,2)
sage: m=M.cuspidal_subspace()
sage: m.rank()
44
sage: m.hecke_bound()
140
}}}
Now we see that the hecke algebra has rank 22 over ZZ
{{{
sage: HA=m.hecke_algebra()
sage: span = [HA.hecke_operator(n) for n in range(1, 140+1) if not
HA.is_anemic() or gcd(n, 29) == 1]
sage: (ZZ^(44^2)).span([i.matrix().list() for i in span])
Free module of degree 1936 and rank 22 over Integer Ring
Echelon basis matrix:
22 x 1936 dense matrix over Rational Field
}}}
So as the comment say's we are unlucky since proj_basis[43] will always be
zero since the hecke algebra has only rank 22
{{{
if proj_basis[dim-1] == 0:
# We got unlucky, choose another projection.
rand_max *= 2
continue
}}}
I suspect there is a problem in that the rank of m is 44 over QQ. But that
the rank of the hecke algebra will be the dimension of m as a complex
vector space so dim should actually be half of what it is now.
I don't know enough of modular symbols yet to know how it should be done
in other weights and other modular groups but I suspect there will be
problems there also.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10510#comment:2>
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