Re: New ellinit interface

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

On Tue, Oct 16, 2012 at 10:38:30PM +0200, Bill Allombert wrote:
> Dear PARI developers,
> 
> Karim and I have been discussing about change to ellinit to make it more useful
> and to replace ellffinit.

So we have decided about a new format for ellinit output (ell object)
I have implemented part of it.
Choice I have made we did not explicitly decide are prefixed by a *

* ell objects will always have 16 components so smallellinit will disappear.
(ellinit(,0) and ellinit(,1) are deprecated and are equivalent to ellinit()).

components 1 to 13 are the same as for smallellinit.
14 is the type, 15 are the type-specific data, 16 are the extended data.
currently my implementation supports the following types:
* t_ELL_Rg: general rings, substitute for smallellinit.
Only basic operations are available.
t_ELL_Q: curves over Q. (Is it easy to find an integral, (possibly non minimal) model ?)
t_ELL_Qp: curves defined over Q, seen over Qp.  (not supported yet)
t_ELL_Fp: curves over F_p
t_ELL_Fq: curves over F_q

Note that there is no curves over R or C. It is almost always better to define such
curves by their periods than by a Weierstrass equation.

ellffinit will be removed.

Reductions:
We frequently want to reduce an elliptic curve over Z to an curve over F_p,
for some p. The simplest way to do that is Ep=ellinit(E,p).

Unfortunately this leads to some pathology:

1) Such reduced curve can be singular. A priori we will not
allow ellinit to define singular curves over F_p.

2) The prime number can be passed implicitly, for example when asked about the
order of a point defined over Fp on a curve defined over Z: e.g.
ellorder(ellinit([0,0,0,1,2]),[Mod(13,17),Mod(11,17)])

3) We still want the ability to compute local invariants at primes of bad reduction.

sometimes both 1,2 and 3) happens. Do we want to allow ellorder on singular curves:
ellorder(ellinit([0,0,0,1,5]),[41,0]*Mod(1,97))

Coordinate changes:
ellchangecurve must be careful to update the type specific and the extended data
correctly.

Cheers,
Bill.