bug in simon_two_descent for elliptic curves

[williamstein]

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

We have

sage: E = EllipticCurve('65a1')
sage: G = E.change_ring(QuadraticField(-56,'a'))
sage: G.simon_two_descent()
(3, 4, [(-9/4 : -3/8*a + 9/8 : 1), (-8/7 : -1/49*a + 4/7 : 1), (1 : 0 : 1), 
  (-6/25*a - 47/25 : 36/125*a - 368/125 : 1), (1/4 : 1/16*a - 1/8 : 1)])

The documentation for simon_two_descent says that the output of Simon 2-descent is

        OUTPUT:
            integer -- "probably" the rank of self
            integer -- the 2-rank of the Selmer group
            list    -- list of independent points on the curve.

Our curve does have rank 3, but the output list above contains five points, so they can't be independent!

Our curve has torsion of order 2, so E(K)/2 E(K) has rank four, so the 3 and four output by Simon descent are right. The only problem is the list, which has too many points in it.

Maybe this is simply a documentation issue, and the docs for simon_two_descent should be changed to say that list is a list of points that generate a subgroup of the MW group of rank r, where r is the first number output by simon_two_descent.

Depends on #13593

Component: elliptic curves

Keywords: simon_two_descent

Author: Chris Wuthrich

Branch/Commit: u/wuthrich/ticket/5153 @ ad2ced2

Reviewer: John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/5153

[JohnCremona]

comment:1

I will email Denis Simon about this, since he has just done another bug fix in ell.gp after I reported a bug to him. Unfortunately the fix he sent works in the latest pari but not with the one Sage uses.

[JohnCremona]

comment:2

Looking at the example in more detail, I see that the points returned do generate a rank 3 group (though I had to use Magma to check the independence), so there are two problems: (1) the points in the list are not independent (even in E(K)/2E(K)) and (2) the numbers 3,4 which are supposed to be lower and upper bounds on the rank are wrong and should be 3,3.

I know how the first happens (since it is doing a descent via 2-isogeny).

This will be possible to fix once I (or someone) have eventually implemented heights of points over number fields, as I promised to do in trac #360 !

[nbruin]

comment:5

Replying to @JohnCremona:

This will be possible to fix once I (or someone) have eventually implemented heights of points over number fields, as I promised to do in trac #360 !

I am not sure that using heights is the best way to get independence if you are doing a 2-descent anyway. The easier thing to do would be to just report back a set of points that minimally generate the 2-selmer group (or at least minimally generate the part of the 2-selmer group generated by the found points)
The only thing you'd miss that way would be if one finds generators for an even index 2 subgroup of the full Mordell-Weil group. I think it's highly unlikely that that would happen.

[williamstein]

comment:6

I am not sure that using heights is the best way to get independence

Another excellent trick I learned from Cremona for getting independence is to use homomorphic images. I.e., reduce to a direct sum of groups of points over a finite field (all tensored with F_p, with p coprime to torsion). Then the rank of the group you have is a bounded below by the dimension of the image.

[JohnCremona]

comment:7

Good point -- the reference is On the computation of Mordell-Weil and 2-Selmer Groups of Elliptic Curves. Rocky Mountain Journal of Mathematics (2002) vol. 32, no. 3, pp.953--967. This is implemeneted over Q in eclib, and it would be possible to do an implementation over number fields. As Mestre points out, this method for testing independence has the great advantage that it does not depend on precision of height computations, only on linear algebra over F_2.