#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 davidloeffler):
I think John's condition "the field of modular functions is generated by
functions whose q-expansions have rational coefficients" is equivalent to
"the subgroup of SL2( Z / N ) corresponding to Gamma is normalized by [1,
0; 0, x] for every x in (Z / N)*". This is at least something we can test
for; but it is not satisfied by many interesting congruence subgroups. (I
don't know about John and Barinder's example; but Elkies' level 9
subgroup, parametrizing elliptic curves with Galois representation that's
surjective mod 3 but not mod 9, certainly does have a canonical model
over Q compatible with its moduli interpretation, but not one which
satisfies this condition).
If we want to do this properly, we should make galois_action take more
data about the modular curve in its arguments, rather than trying to make
the answer be valid for all modular curves simultaneously, which is
impossible.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13805#comment:5>
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