Re: Heights over number fields

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Thu, Oct 01, 2009 at 05:15:07PM -0500, Ariel Pacetti wrote:
>
>
> This is a more general question than the application I have in mind. I  
> just want to know is some points are linearly independent or not.

Maybe I misunderstand your problem, but I would embed the number field in C
and use the elliptic logarithm (ellpointtoz in GP term).

The following show how it can be done:

\p180
dlog(E,P,G)=
{
  local(zP,zG,om,l);
  zP=ellpointtoz(E,P);
  zG=ellpointtoz(E,G);
  om=E.omega;
  l=lindep([zP,zG,om[1],om[2]]);
  -l[2]/l[1]
}

p=nextprime(2^160);
a=random(2^160)
E=ellinit([0, 0, 1, -1, 0]);
P=[0.,0.];
Q=ellpow(E,P,a);
dlog(E,Q,P)

Cheers,
Bill.