Re: Support for elliptic curves over number fields

[John Cremona <john.cremona@gmail.com>]

On 16 July 2016 at 19:04, Bill Allombert
<Bill.Allombert@math.u-bordeaux1.fr> wrote:
> On Sat, Jul 16, 2016 at 06:06:55PM +0200, Bill Allombert wrote:
>> Dear PARI developers,
>>
>> We have added basic support for elliptic curves over number fields and L
>> function of elliptic curves over number fields.
>>
>> This is an example:
>>
>> ? N=nfinit(a^3-26);
>> ? E=ellinit([a,0,1,0,0],N);
>> ? lfun(E,1)
>> %4 = 0.24961216776576924744553489082015201012
>
> For thus who are interested, the following script checks the
> Birch and Swinnerton-Dyer conjecture for rank-0 curves.
>
> ? N=nfinit(a^3-26);
> ? E=ellinit([a,0,1,0,0],N);
> ? bsd(E)
> %3 = 0.99999999999999999999999999999999999999
>
> (I do not know how to compute the height of points for rank-1 curves.)

I could help with that.  The local heights at finite primes are easy,
while at the infinite places my student Marco Caselli has a preprint
showing how the Mestre AGM method can be used for complex as well as
real places.  It is nearly ready, and when it is, it will be easy to
add efficient computation of canonical heights over arbitrary numbers
fields to pari.  (One could do it already, using the Silverman-Tate
algorithm as implemented in Magma and Sage, but that does not behave
well numerically.)

John

>
> Cheers,
> Bill.