#5080: Bug in decomposing modular symbol subspace
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Reporter: robertwb | Owner: craigcitro
Type: defect | Status: new
Priority: major | Milestone: sage-4.0.1
Component: modular forms | Keywords:
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Comment(by davidloeffler):
I have had a careful look at this, and I think I know what's going on. The
problem is that for each of these curves, if f is the corresponding
newform, then there is a finite set of forms f_1 ... f_m (none of them
equal to f) in the space such that for every p, a_p(f) = a_p(f_i) for some
i. It was a bit of a surprise to me that this is possible, but it doesn't
contradict multiplicity one, and in fact if you take any fixed form and
consider its twists by chi1, chi2, and chi1 * chi2 for any two quadratic
characters chi1, chi2 of coprime moduli then you get an example.
This mightily confuses two functions for submodules of Hecke modules:
"complement" and "dual_free_module". The former has a workaround, in that
if it can't find a complement using only one Hecke operator at a time, it
falls back on calling "decomposition" (which is slower, but is immune to
this problem) and works out the complement using that. The latter doesn't.
But anyway, the two are basically doing the same thing, since the embedded
dual free module of a submodule V is by definition the annihilator of the
Hecke-stable complement of V (when this exists). So the fix is to get rid
of the existing "dual_free_module" routine and replace it with a simpler
routine that calls "complement" and then does some trivial linear algebra.
I will post a patch when I get a chance to code it up.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5080#comment:2>
Sage <http://sagemath.org/>
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