tighter upper bound on elliptic curve rank

[R. Andrew Ohana]

[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.

[R. Andrew Ohana]

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 :)

[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.

[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 :).

[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.

[wuthrich]

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:

​9a069c7

Trac #13593: ranks of elliptic curves over number fields

[wuthrich]

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

[git]

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

​b1550dc

Trac 13593: Do not put torsion points into gens.

[wuthrich]

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

[Frédéric Chapoton]

There is a typo "not imcluded"

[git]

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

​bf7617d

spelling error

[John Cremona]

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