|Bill Allombert on Sat, 20 Oct 2012 17:03:01 +0200|
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|Re: New ellinit interface|
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.