John Cremona on Tue, 24 Jun 2014 15:13:26 +0200

 Re: ellheegner

On 24 June 2014 13:44, Ariel Martin Pacetti <apacetti@dm.uba.ar> wrote:
>
> 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?).

ellheegner is in 2.7.1 which is on its way into Sage (see
http://trac.sagemath.org/ticket/15767).  When that is done I'll add an
interface to it so that for E an alliptic curve over Q of rank 1,
E.heegner_point() will return the point which ellheegner computes.  I
have been using this (as Bill knows) for 2 or 3 years already, and in
fact a large proportion of the rank one generators in my tables for
curves of conductor > 130000 were found using this.

>
> 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).

In that case something simpler than Bill's excellent implementation
might suffice.  The attached is very basic and was used as a demo by
me years ago (at the Arizona Winter School in 1999 I think).  It is
possible that the number of coefficients used (40000) was chosen as
being the maximum length of a gp array at the time!

John

>