Re: New GP function ellrank (2-descent)

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Feb 05, 2021 at 04:49:28PM +0000, John Cremona wrote:
> Bill,
> 
> This is great news!  When mwrank is completely redundant then I can
> really retire.
> 
> Some questions:
> 
> 1. When you show that the output is [1,1,[]] you are asserting that
> the rank is exactly 1 even though you do not (yet) have any points.
> Are you using the theorem that (analytic rank=1) => (rank=1)?
> 2. Do you also use (analytic rank=0) => (rank=0)? In both these cases,
> is the determination of the analytic rank rigorous?

ellrank does not use analytic method, because the conductor is usually
much too large.

I am really not an expert on 2-descent but how I understand it is:

- the rank of the 2-selmer is equal to the sum of the 2-rank of the torsion group,
the rank of the curve, and the 2-rank of the Tate Shafarevich group.

- the 2-rank of the Tate Shafarevich group is even.

Assuming the above, then the parity of the rank is known.
In the example, we conclude that the rank is odd and <= 2 so it must be
1.

(Another way is to assume that the root number is equal to (-1)^rank, which
as I understand was proved by Elkies at least in some case).

> 3. Is the list of points always independent (modulo torsion)?  Denis's
> output sometimes included torsion and/or included dependent points (if
> I recall correctly).

Denis script always include the 2-torsions points.
For simplicty I removed them, because it was confusing.
The remaining points are always independent.

> 4. For analytic rank 1 curves do you use ellheegner() instead of
> descent?  I hope so since ellheegner is totally and utterly brilliant! 

Unfortunately, ellheegner only works for curve with small conductor, so
it is not used.

Note that Mark Watkins has published an algorithm which use 4-descent
to speed up the Heegner point method. Unfortunately I do not know
how to do 4-descent.

- - - -

I implemented saturation but I do not know how to interface it with
ellrank, in particular how to chose the bound.
It seems the bound computed by mwrank is often impractical, so mwrank
use 1000.

Did you see the paper https://arxiv.org/pdf/2007.09514.pdf ?
Can it be used to improve the bound ?

Cheers,
Bill