Re: ellheegner

[Ariel Martin Pacetti <apacetti@dm.uba.ar>]

Dear Bill,

> Hello Ariel,
> Could you provide an example ?

Take the elliptic curve 37a1 (which has rank 1, and prime conductor). In 
this case, take any d such that kronecker(-d,37)=1, and construct a 
Heegner point attached to d (for general N, you need each prime dividing N 
to split in Q[\sqrt{-d}] if you take the whole ring of integers):

d= -3 --> P=[-1,0]
d=-7 --> P=[0,0]
...
d=-27 --> P=[2,-3]
d=-4*33 --> P=[-1,0] (here the class number is not one, so you need to 
take the trace I was talking about)

> This can be done with minimal change to the PARI source code.
>
> Hwoever, PARI definition of the Heegner point is
> "A  non-torsion  rational  point  on  the curve,  whose canonical height is equal to the
> product of the elliptic regulator by the analytic Sha"

If I am not mistaken, then this implies that the point is a generator of 
the free part. Then you do not have much choice (up to torsion), but in 
general Heegner points give you a multiple of such element. I guess that 
you are computing some Heegner point, and use it to search for a generator 
(you have few choices), so it is a little tricky to call this routine 
"heegnerpoint".

Regarding John's comment, what heegnepoint is doing is much better, and 
much faster! (I coputed the generator in SAGE and took many minutes, while 
in GP it was done in a few seconds). I saw that ellheegner is not part of 
Pari's version in SAGE, is there any reason? (or just that it is in the 
unstable version only?).

I am finishing teaching a course on elliptic curves, and am planning to 
give a computer based class, which was the main motivation for the exact 
implementation of Heegner points (which was discussed durying the course).

Thanks for the answers!

Ariel