John Cremona on Tue, 24 Jun 2014 15:13:26 +0200 |
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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 > > Thanks for the answers! > > Ariel >
Attachment:
heeg_demo.gp
Description: application/gnuplot