Opened 10 years ago

Closed 9 years ago

#13593 closed enhancement (fixed)

tighter upper bound on elliptic curve rank

Reported by: R. Andrew Ohana Owned by: John Cremona
Priority: major Milestone: sage-6.1
Component: elliptic curves Keywords:
Cc: Alyson Deines Merged in:
Authors: R. Andrew Ohana, Chris Wuthrich Reviewers: John Cremona
Report Upstream: N/A Work issues:
Branch: u/wuthrich/ticket/13593 (Commits, GitHub, GitLab) Commit: bf7617dbf1778300f421ca0e0305d881a0c8bda2
Dependencies: Stopgaps:

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Description (last modified by John Cremona)

[See #15608 for a list of open simon_two_descent tickets]

The current upper bound in rank_bounds do not take into account two torsion. This corrects this and thus gives a tighter upper bound.

Attachments (1)

trac13593.patch (833 bytes) - added by R. Andrew Ohana 10 years ago.

Download all attachments as: .zip

Change History (17)

Changed 10 years ago by R. Andrew Ohana

Attachment: trac13593.patch added

comment:1 Changed 10 years ago by R. Andrew Ohana

Description: modified (diff)
Status: newneeds_review

comment:2 Changed 10 years ago by R. Andrew Ohana

Status: needs_reviewneeds_work

need to fix doctests...

Update: well having trouble finding a new example where the bounds disagree -- guess this is what I get for trying to improve something :)

Last edited 10 years ago by R. Andrew Ohana (previous) (diff)

comment:3 in reply to:  2 Changed 10 years ago by John Cremona

Replying to ohanar:

need to fix doctests...

Update: well having trouble finding a new example where the bounds disagree -- guess this is what I get for trying to improve something :)

I do think that Simon's scripts already take 2-torsion into account, even though he returns 2-torsion points in the list of gens (which I remember filtering out). Needs checking though.

comment:4 Changed 10 years ago by R. Andrew Ohana

If I recall, the upper bound is supposed to be a bound on the rank of the 2-Selmer group, which would not take into account 2-torsion -- unless I'm being totally stupid, which could very well be the case considering how new I am to this stuff :).

Last edited 10 years ago by R. Andrew Ohana (previous) (diff)

comment:5 Changed 9 years ago by Jeroen Demeyer

Milestone: sage-5.11sage-5.12

comment:6 Changed 9 years ago by wuthrich

A simple example:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([2+3*i,0])
sage: E.simon_two_descent()
(0, 1, [(0 : 0 : 1), (0 : 0 : 1)])
sage: E.rank_bounds()
(0, 1)
sage: Em = magma(E)
sage: Em.TwoSelmerGroup()
Abelian Group isomorphic to Z/2
...

Simon's script returns indeed as the second argument the dimension of the 2-Selmer group and currently rank_bounds copies that. So the supposed change here is indeed good. However, we need to add an example in the doctest, too.

Furthermore, I will correct a few other things there, while I am at it.

comment:7 Changed 9 years ago by wuthrich

Branch: u/wuthrich/ticket/13593
Created: Oct 11, 2012, 11:28:04 PMOct 11, 2012, 11:28:04 PM
Modified: Dec 28, 2013, 12:56:43 AMDec 28, 2013, 12:56:43 AM

comment:8 Changed 9 years ago by wuthrich

Authors: R. Andrew OhanaR. Andrew Ohana, Chris Wuthrich
Commit: 9a069c7225480773cd6429e87f10bf5d0d8855e0
Status: needs_workneeds_review

I also corrected the documentation - it was quite bad ! Meaning mathematically incorrect. Hope it is better now.

At some point we will want to filter out generators for E(K)/2E(K) - or even E(K). However, I left for now to return all points that Simon's script finds.


New commits:

9a069c7Trac #13593: ranks of elliptic curves over number fields

comment:9 Changed 9 years ago by wuthrich

Description: modified (diff)
Priority: minormajor

... and then I spotted that the list may contain points of order 2. the second commit filters the torsion points out of the list.

comment:10 Changed 9 years ago by git

Commit: 9a069c7225480773cd6429e87f10bf5d0d8855e0b1550dc0081c5d6cb971af025108cacf653b5fa8

Branch pushed to git repo; I updated commit sha1. New commits:

b1550dcTrac 13593: Do not put torsion points into gens.

comment:11 Changed 9 years ago by wuthrich

The proposed resolution of #5153 is a follow up on this one here.

comment:12 Changed 9 years ago by Frédéric Chapoton

There is a typo "not imcluded"

comment:13 Changed 9 years ago by git

Commit: b1550dc0081c5d6cb971af025108cacf653b5fa8bf7617dbf1778300f421ca0e0305d881a0c8bda2

Branch pushed to git repo; I updated commit sha1. New commits:

bf7617dspelling error

comment:14 Changed 9 years ago by John Cremona

Description: modified (diff)

comment:15 Changed 9 years ago by John Cremona

Reviewers: John Cremona
Status: needs_reviewpositive_review

I looked at all three commits and approve of what they do. Test pass -- positive review.

comment:16 Changed 9 years ago by Volker Braun

Resolution: fixed
Status: positive_reviewclosed
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