#10735: Simon 2-descent only returns an upper bound on the 2-Selmer rank
-------------------------------------------------+-------------------------
Reporter: weigandt | Owner: cremona
Type: defect | Status: new
Priority: major | Milestone: sage-6.2
Component: elliptic curves | Resolution:
Keywords: simon_two_descent | Merged in:
Authors: | Reviewers:
Report Upstream: Reported upstream. No | Work issues:
feedback yet. | Commit:
Branch: | Stopgaps:
Dependencies: #11005 |
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Old description:
> [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.
>
> This curve definitely possesses 2-covers which are everywhere locally
> soluble EXCEPT at that infinite place. These probably explain the
> discrepancy.
>
> {{{
> sage: E=EllipticCurve('438e1')
> sage: E.simon_two_descent()
> (0, 3, [(13 : -7 : 1)])
> sage: E.selmer_rank() #uses mwrank
> 1
> sage: E.sha().an()
> 1
> }}}
New description:
[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.
{{{
sage: E=EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [(13 : -7 : 1)])
sage: E.selmer_rank() #uses mwrank
1
sage: E.sha().an()
1
}}}
The explanation for this is that `E.simon_two_descent()`, unlike Cremona's
`mwrank`, does not do a second descent and therefore only determines an
upper bound on the 2-Selmer rank.
--
Comment (by pbruin):
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 ะจ, or doesn't it?
--
Ticket URL: <http://trac.sagemath.org/ticket/10735#comment:10>
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