#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 | Keywords: cusps galois
Work issues: | Report Upstream: N/A
Reviewers: | Authors:
Merged in: | Dependencies:
Stopgaps: |
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This is a follow-up from #13253, sicne that ticket did not completly fix
the problem. To quote from a late comment there:
I was using this function recently and unfortunately the documentation
makes incorrect claims for its applicability! Being based on Steven's
book, it works for *some* congruence subgroups of level N, but *not all*
of them.
The example I have is this. I have a subgroup of level 13, index 91,
consisting of matrices in PSL(2,Z) whose mod-13 reduction lie in a
subgroup of PSL(2,13) isomorphic to A4. Under the action of A4 the 84
cusps of Gamma(13) form 7 orbits of size 12 each. But the action of (t mod
13) is only well-defined for t=1,5,8,12 (i.e. the cubes mod 13). I could
give examples of 2 cusps c1,c2 which are A4-equivalent but the results of
c.galois_action(2,13) for c=c1,c2 are not A4-equivalent. This can be
explained by looking carefully at Stevens' proof of his proposition, which
relies on the field of modular functions for the group in question being
generated by by functions whose Q-expansions have rational coefficients.
This is true for Gamma(N), Gamma0(N), Gamma1(N), but not in general. In my
case the field of coefficients required is the cubic subfield of
Q(zeta13), which explains why Stevens's formula is only valid when t is a
cube.
I think that the way to fix this is to change the documentation so that
the function does not claim to do more than it does. A complete fix would
require something really new, with input more than just the level N and an
invertible residue class t mod N. I was able to work out my specific
example, but a general implementation would be quite hard. [For the
record, the 7 cusps consist of 2 Galois orbits, of size 3 and 4. Using
Sage's function restricted to t=5 (which generates the cubes mod 13) I
found a 4-cycle and 3 fixed points, including infinity, and as I already
knew that infinity was in an orbit of size 3 that was sufficient!]
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13805>
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