John Cremona on Mon, 11 Dec 2023 15:02:53 +0100

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Re: question on the use of Weber's Functions

On Mon, 11 Dec 2023 at 13:03, Bill Allombert <> 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
> 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

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).