#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 mderickx):
Hi, I think this error slipt in because I'm more used to thinking
something that I will denote by X'(N). Namely the modular curve that
parametrized Z/NZxZ/NZ immersions. The action of galois is trivial on
Z/NZxZ/NZ there are no problems in this case since for a congruence
subgroup Gamma \supset Gamma(N) one can define the quotient X'(N)/(Gamma/
Gamma(N)) and in this way the galois action on the cusps is compatible
between all modular curves.
This X'(N) has some disadvantages, namely its not geometrically connected
and hence not a model of H^*/Gamma(N), but instead it's a model of several
copies of H^*/Gamma(N). What is used in sage is the modular curve X(N)
that parametrizes mu_NxZ/NZ immersions together with some condition on the
weil paring. This X(N) this will be an irreducible component of my X'(N)
after base change to Q(zeta_N)
Now in sage I just didn't think about the fact that the action of
GL_2(Z/NZ) cannot be defined on mu_NxZ/NZ and hence that it is not
possible to define an arbitrary modular curve of level N as X(N)/(Gamma/
Gamma(N)) since this will mess up the galois action as John mentioned.
I wonder whether the condition mentioned by John that:
The field of modular functions for the group in question is generated
by functions whose Q-expansions have rational coefficients.
Is equivalent with the condition:
H/Gamma can be given a model over Q such that X(N)=H/Gamma(N) ->
H/Gamma is defined over Q.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13805#comment:4>
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