#8451: improve galois representation for elliptic curves
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Reporter: wuthrich | Owner: cremona
Type: defect | Status: needs_work
Priority: major | Milestone: sage-4.4
Component: elliptic curves | Keywords: elliptic curves, galois
representation, is_surjective
Author: Chris Wuthrich | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by was):
* status: needs_review => needs_work
Comment:
The following is from Drew:
"Hi William,
I spent some time this afternoon testing the new Galois representation
code using the test version of Sage that you set up.
I tried out several curves with interesting Galois images and was able to
find a number of potential issues in the implementation, and one
definitive bug. In several cases the image_type function output the
message:
"The image could not be determined, it is likely that is a bug" (*)
I did not have time to try and trace these through the code, but I have
listed some examples that will elicit this message below. I suspect that
someone familiar with the code can track these down more quickly than I.
The only primes where I found problems were 5 and 7.
For 5, below are two different examples that elicited (*):
Curve Mod 5 image
[0,0,1,-25650,1570826] Index 2 subgroup of normalizer of split Cartan
[1,-1,1,-5,2] Normalizer of non-split Cartan
For the curve [0,0,0,-56,4848] the image_type function reported that it
could not determine the image but that it is likely to be contained in the
normalizer of a non-split Cartan. This is probably harmless, but I note
that the mod 5 image of this curve has order 32 and is the normalizer of a
*split* Cartan.
For 7 the following examples elicited (*)
Curve Mod 7 image
[1,-1,1,-2680,-50053] Index 4 subgroup of normalizer of split Cartan
[1,-1,0,-107,-379] Index 2 subgroup of normalizer of split Cartan
More seriously, the image_type function mis-identified the mod 7 image of
the curve [0,0,1,2580,549326]. It claims that the image is contained in
the normalizer of a non-split Cartan, but this is not true, it lies in the
normalizer of a split Cartan. As a proof that it is *not* in the
normalizer of a non-split Cartan, I note that for p=19, the factorization
pattern of Phi_7(X,j(E)) is 1,1,6, which implies that the mod 7 image of
Frobenius has an eigenvalue and is of order 6 in PGL(2,F_7). But no
element of the normalizer of a non-split Cartan has this property.
Aside from checking these particular exceptional cases, I think it would
be very worthwhile to do an automated run through some or all of the
Cremona or Stein-Watkins tables and comparing the results of
Galois_representation and galrep (at least as to surjectivity). I suspect
this is more likely to find problems than further manual testing or code
review.
Regards,
Drew"
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8451#comment:11>
Sage <http://www.sagemath.org>
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