#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-5.13
Component: modular forms | Resolution:
Keywords: cusps galois | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by pbruin):
It is certainly true that the Galois action depends on more than just the
element of '''P'''^1^('''Q'''). In fact, it depends on a choice that is
usually only made implicitly. Namely, a modular curve ''X,,H,,'' of level
''n'' is defined by a subgroup ''H'' of GL,,2,,('''Z'''/''n'''''Z''').
The natural field of definition of ''X,,H,,'' is ''K,,H,,'' =
'''Q'''(ζ,,n,,)^det ''H''^. To get a modular curve defined over '''Q''',
we therefore need the determinant to be surjective. Let ''H'',,1,, be the
group of matrices of determinant 1 in ''H''. When starting from a
congruence subgroup, one gets ''H'',,1,, instead of ''H''; one needs to
specify how to extend this to a subgroup ''H'' as above.
For Γ,,1,,(n), the ''H'' that one needs to take to get the usual modular
curve ''X'',,1,,(''n'') is the subgroup of GL,,2,,('''Z'''/''n'''''Z''')
consisting of the matrices with bottom row (0 1). In the context of
embeddings of ('''Z'''/''n'''''Z''')^2^ into elliptic curves, this
corresponds to marking the point of order ''n'' which is the image of the
second basis vector.
For general ''H'', not necessarily with surjective determinant, there is
an action of ''G,,H,,'' = Gal('''Q'''(ζ,,n,,)/''K,,H,,'') =
('''Z'''/''n'''''Z''')^×^ on the set of cusps. The recipe for computing
it is as follows, if my back-of-the-envelope calculation is correct.
Let ''s'' be in ''G,,H,,'' and let ''a''/''b'' be in '''P'''^1^('''Q''').
Choose a matrix ''g'' in ''H'' of determinant ''s''^-1^ in ''G,,H,,''.
Map ''a''/''b'' in the obvious way to a primitive column vector ''v'' over
'''Z'''/''n'''''Z'''. Compute ''w'' = ''g''(''sv''); this is still a
primitive column vector. Lift ''w'' to a primitive column vector ''x''
over '''Z'''. Finally, map ''x'' in the obvious way to an element of
'''P'''^1^('''Q'''). This represents the desired cusp obtained by
applying the Galois automorphism corresponding to ''s'' to the cusp
''a''/''b''.
In the case of the usual congruence subgroups, one can take ''g'' to be
the diagonal matrix with (''s''^-1^ 1) on the diagonal. This is
consistent with David's remark above. For a general congruence subgroup,
one first has to extend its image in SL,,2,,('''Z'''/''n'''''Z''') to a
group ''H'' as above (with appropriate image of the determinant, possibly
giving a larger base field than '''Q'''). Second, one has to implement
choosing the element ''g'' as above.
To implement this in complete generality, we would have to allow the user
to construct the modular group by giving a subgroup ''H'' of
GL,,2,,('''Z'''/''n'''''Z'''), not just a congruence subgroup.
--
Ticket URL: <http://trac.sagemath.org/ticket/13805#comment:9>
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