Re: question on the use of Weber's Functions

On Mon, 11 Dec 2023 at 13:03, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:

On Sun, Dec 10, 2023 at 07:55:09PM -0800, American Citizen wrote:
> Hello:
>
> I obtained from Andre Robatino back in the mid-1990's an elegant GP-Pari
> script which I modified to find the mock Heegner points for rank 1 Congruent
> Number elliptic curves. I used his script to find the MW groups for all but
> 3 rank=2 curves for the first 1,000,000 elliptic curves.
>
> The ingenious part of the script which Andre created uses Weber's functions
> and has an almost quadratic convergence to the non-torsion rational point on
> the curve. For example, I believe it took only 6 passes for me to find the
> Mordell Weil generator for a point of height 40593.31146980... which is very
> high on the rank=1 curve of n = 958957.
>
> Has anyone here used Weber's functions to help find the rational points for
> the Mordell_Weil generators on the general rank=1 curve E(Q) ?
>
> See https://dl.acm.org/doi/book/10.5555/922720
>
> Andre gave me a copy of his master's thesis.
>
> I am very intrigued that these Weber functions can possibly make a break
> through in finding the MW group (at least on rank=1 curves) in a much faster
> way than using Heegner points for general rank=1 curves.

This is interesting. I know about the mock Heegner points method , but what I
had seen did not seem practical, so I would be very interested in this GP
script!

Do you really mean "mock" Heegner points, Randall? I don't think that they had been invented in the 1990s. (They use p-adic methods developed by Marc NMasdeu and others, and are only conjectural.)

I know that ellheegner is not optimal for CM curves, and congruent numbers curves
are all CM, so maybe this is the reason ?

Certainly for these curves computing the a_p is faster than by pint-counting, as there is a simple formula (due to Gauss, and a_p=0 for all p=3 mod 4, etc).

John

Cheers,
Bill