Bill Allombert on Mon, 11 Dec 2023 14:03:42 +0100 |
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Re: question on the use of Weber's Functions |
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! I know that ellheegner is not optimal for CM curves, and congruent numbers curves are all CM, so maybe this is the reason ? Cheers, Bill