Re: New GP function ellrank (2-descent)

[John Cremona <john.cremona@gmail.com>]

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

John

On Fri, 5 Feb 2021 at 16:11, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> Dear PARI developers,
>
> I have added a new GP function 'ellrank' to the master branch.
> This is a port of Denis Simon GP script ellQ.gp.
> However the interface is different, it returns [r,R,V] where r is a
> lower bound for the rank, R is an upper bound and V is a list of point.
> If the full gorup is determined, r==R==#V.
>
> ? E = ellinit([-157^2,0]);
> ? ellrank(E)
> %2 = [1,1,[]]
> The rank is 1 but no points was found.
> It is possible to ask ellrank to try harder
> ? ellrank(E,10)
> %5 =
> [1,1,[[-43565582610691407250551997/609760250665615167250729,562653616877773225244609387368307126580/476144382506163554005382044222449067]]]
>
> It should compare favorably to John Cremona mwrank (which use a
> different algorithm), except it does not do saturation yet, but this
> will be added soon.
>
> Finally, I like to thanks Denis Simon for improving ellQ.gp over all
> this years.
>
> Cheers,
> Bill
>