Re: question on mapping points from an elliptic curve back to a quartic

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

On Thu, Oct 10, 2024 at 11:50:04AM -0700, American Citizen wrote:
> Okay Bill and John:
> 
> BIll, thank you for your short explain, this is the first time that I
> actually understood what is going on.
> 
> John, I did read your book, but did NOT understand this part until now.
> 
> and to all, I actually started with the quartic, not the elliptic curve,
> there were two GP-Pari commands available which found E, quartic_to_ellmap()
> and ellfromeqn(). I used the first command to find E, since I also needed
> the forward and inverse maps for the points.
> 
> I did NOT know that Q was a two-cover for E.
> 
> Does this mean that any E found for a given Q forces (better word choice??)
> the Q to be a 2-cover when any GP-Pari command discovering E is used on the
> quartic?

E is called the Jacobian curve of the genus-1 quartic Q, and yes Q will be
a 2-cover of E. 

> Can I safely assume that I have to multiply the points on E by 2, before
> mapping back to Q? That's what I guess from what has been shared here from
> Bill

As John pointed out, I made a simplification in my post.
In general, an elliptic curve admit several 2-covers which correspond to
cosets in E/[2]E.
So the set of points of E that map to Q will be of the form { P_Q + 2*A, A in
E(\Q) } for a _fixed_ P_Q that depends on Q. These sets form a partition of
E(\Q), so if you have only one cover you can take P_Q to be the point at infinity.
Otherwise P_Q can be taken as a sum of some of the generators and a point
of 4- or 8-torsion if any.

Cheers,
Bill.