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Simon 2-descent only returns an upper bound on the 2-Selmer rank #10735
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comment:2
This is a bug in Simon's script indeed. I have emailed him about this one, too, as it happens with the later version of his file in gp, too. |
Upstream: Reported upstream. No feedback yet. |
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Dependencies: #11005 |
comment:7
It seems to me that the bug is not caused by failing to detect non-solubility at the real place. In fact, Simon's script computes the same 2-isogeny Selmer ranks as Output of
Output of
The 3 in the last line, which should be the rank of the 2-Selmer group according to the documentation, is the result of computing (rank of |
comment:8
That is indeed bad. The fomula (after the first descent) is indeed dim Sel_phi + dim Sel_phihat + dim E[2] - 2 . this is a upper bound to dim Sel_2 and they have the same parity. But they need not be equal as the second descent in mwrank finds. The difference is a subquotient of Sha(E'). Conclusion: The output of Simon's algorithm is an upper bound on the 2-Selmer group, which is correct in parity, but not necessarily equal. In propose that we change the documentation, for I don't image that Simon's script could give back to full answer - though I have not looked at it. |
comment:9
You are right. DS does no second descent (unlike mwrank when over Q and with 2-torsion) and the second descent can give a better upper bound on the rank. |
comment:10
Changing the documentation does indeed sound like the right solution here. The correctness of the parity of this upper bound probably relies on finiteness of Ш, or doesn't it? |
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comment:11
No the correctness of the parity is unconditional. (This is used for instance in the proof of the p-parity conjecture via p-isogeny.) |
Author: Peter Bruin |
comment:12
Here is a patch for the documentation of There are a few other improvements to the docstring. In particular, it contained several calls to |
Commit: |
comment:13
Another thing: the method |
comment:14
We removed the points of finite order from Now to this ticket. I have run all tests and I am certainly happy with the improvement. So I give a positive review. I am not sure about the random seed issue above, but I trust you. Otherwise someone will complain at some point. |
Reviewer: Chris Wuthrich |
Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. Last 10 new commits:
|
comment:16
There was just a trivial merge conflict with #9322, which also has positive review, so I merged that branch and made it a dependency. |
Changed branch from u/pbruin/10735-simon_two_descent_doc to |
[See #15608 for a list of open simon_two_descent tickets]
Given an elliptic curve E the method E.simon_two_descent() returns an ordered triple. This consists of a lower bound on the Mordell-Weil rank of E, an integer which is supposed to be the F_2 dimension of the 2-Selmer group of E, and list of points, generating the part of the Mordell-Weil group that has been found.
Sometimes the second entry is larger than the actual 2-Selmer rank as computed by mwrank, and predicted by BSD. The first curve I know of for which this happens is the elliptic curve '438e1' from Cremona's tables.
The explanation for this is that
E.simon_two_descent()
, unlike Cremona'smwrank
, does not do a second descent and therefore only determines an upper bound on the 2-Selmer rank.Depends on #11005
Depends on #9322
Upstream: Reported upstream. No feedback yet.
CC: @JohnCremona @williamstein @rlmill
Component: elliptic curves
Keywords: simon_two_descent
Author: Peter Bruin
Branch/Commit:
732191d
Reviewer: Chris Wuthrich
Issue created by migration from https://trac.sagemath.org/ticket/10735
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