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Local data of elliptic curves should not do any global work #11630
Comments
comment:1
That is a sensible suggestion, and if you write a patch I'll review it! Ideally I would like the code to say "if you know that P is principal and have a generator, the use it, otherwise...", but I suppose that is what the uniformizer function might do anyway. I think that even if we did have a version for Tate's algorithm which applied directly to an elliptic curve over a local field, I would still want a global version -- which used generators of prime ideals for fields of class number 1 at least! |
Author: Chris Wuthrich |
comment:2
This patch does what I suggested + it also changes global_minimal_model. This function was only implemented when the class number was 1, now it tries to change to a global minimal model in any field. The patch should be applied after #11540. |
Reviewer: John Cremona |
comment:3
First a technical point: the patch does not apply to 4.7.1.rc1+#11540. Since 4.7.1 is almost out could you rebase the patch to the latest release candidate? I'll test more after that. Meanwhile I am not sure that the logic is correct in the new, more general, global_minimal_model() function. Are you assuming that the is a global minimal modell iff after replacing with a local minimal model at every relevant prime, one after the other, the result is minimal at all primes? I'm sure that is not right. On my (infinitely long) to-do list is a function that determines whether there is a global minimal model for any curve, which comes down to computing a certain ideal and testing if it is principal; if so then (using a generator for that ideal) one can find the minimal model. That should be done, but not on this ticket. |
Work Issues: rebase to 4.7.1 |
comment:4
Thanks, John, for the quick review and for spotting my stupid error. In fact all I really wanted was that global minimal models are recognised when they are, but that was a bad idea. I agree with your comments and with your suggestion to open another ticket for this. This is ticket #11697 . I will rebase it today, hopefully, and delete the relevant passages. (Sorry, but I have decided quite a while ago not to work with intermediate versions anymore, but now 4.7.1 is here.) |
comment:5
Ok. I rebased it and I removed the changes in |
comment:6
The patch still applies to 4.7.2. |
Changed reviewer from John Cremona to John Cremona, PatchBot |
comment:7
The patch does not apply to 4.8 or to 5.0.beta7, according to the patchbot. |
Changed work issues from rebase to 4.7.1 to rebase to 5.0 |
exported against 5.9 |
comment:9
Attachment: trac_11630_localize_tates_algorithm.patch.gz I have removed the whitespaces. This patch should apply to 5.9 + #12509. |
Branch: u/wuthrich/ticket/11630 |
Commit: |
comment:13
I have put this onto my to-do list to attempt my first git-style review, but in view of the season do not promise to do it in the next few days. |
comment:14
I am giving this a positive review despite the patchbot's red blob. |
Changed reviewer from John Cremona, PatchBot to John Cremona |
Changed work issues from rebase to 5.0 to none |
... unless asked to do so.
I am talking about the line 687 in sage.scheme.elliptic_curves_ell_local_data which reads
While it can be useful, especially when one wants a global minimal model, it is often very harmful. If the class group of the field is huge or difficult to compute, one will not be able to determine the fibres of the Neron model, simply because of this line.
One should add a switch which is by default set to use the second case and only if needed to the first case.
Of course, in an ideal world Tate's algorithm should be implemented for any elliptic curve over a local field, rather than a number field.
CC: @JohnCremona
Component: elliptic curves
Keywords: Tate's algorithm
Author: Chris Wuthrich
Branch/Commit: u/wuthrich/ticket/11630 @
d2d3457
Reviewer: John Cremona
Issue created by migration from https://trac.sagemath.org/ticket/11630
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