#13805: galois_action on cusps has a bug and incorrect documentation II
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Reporter: cremona | Owner: davidloeffler
Type: defect | Status: new
Priority: major | Milestone: sage-5.6
Component: modular forms | Resolution:
Keywords: cusps galois | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by mstreng):
Hi John,
Yes, you are right about that. All this assumed that the function field
was generated by the field of functions with Fourier coefficients in QQ,
and even used that as the definition of the rational model. So it does not
directly apply to your example. However, on a quick reading, I can't find
where the documentation claims that it does apply to all modular curves.
In any case, you are right that a warning in the documentation is in
order. Something like "This only gives well-defined cusps if the field of
functions is generated by functions with rational Fourier expansions, such
as for Gamma = ...."?
At the same time, maybe the output is not completely bogus even in your
example. Does the following interpretation make sense?
Your curve has a model over a cubic field, and Gal(QQab / QQ) acts by
sending cusps of your curve to cusps of Galois conjugates of your curve.
So even though (2 mod 13)*c1 is not equivalent to (2 mod 13)*c2 modulo
A_4, it is equivalent modulo some conjugate of A_4.
Marco
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13805#comment:1>
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