From charlotte.griffin@new.oxford.ac.uk Thu Apr 13 11:28:06 2006
Date: Thu, 13 Apr 2006 11:28:06 +0100
From: Charlotte Griffin <charlotte.griffin@new.oxford.ac.uk>
To: flynn@maths.ox.ac.uk
Subject: elliptic curves

Victor,

I'm stuck on one point of the elliptic curves notes, I wonder if you could 
explain what I'm missing:

The very first line on page 17 - that the point at infinity maps to (0,0).  
I don't see why.

I'm sure it's something obvious and I'll kick myself when I find out.

Thank you, and Happy Easter,

Charlotte

From charlotte.griffin@new.oxford.ac.uk Thu Apr 13 13:00:49 2006
Date: Thu, 13 Apr 2006 13:00:49 +0100
From: Charlotte Griffin <charlotte.griffin@new.oxford.ac.uk>
To: Victor Flynn <flynn@maths.ox.ac.uk>
Subject: Re: elliptic curves

Victor,

Thank you, I think I understand now, except I'm a bit confused by the 
following:

Any birational map given as (x,y) -> (z,w) is in fact shorthand for
(X,Y,S) -> (z,w,1).
So, while the map is defined for the point at infinity, the point at infinity 
is not in the image of the map.

Is that right?

Charlotte


> Dear Charlotte,
> 
>    As always, all of the curves should be understood to be projective
> (the affine form is just a shorthand notation for the projective curve). 
> To understand what happens to the point at infinity, write the
> curve in projective form. Recall that
> E : y^2 = x^3 + A x + B
> is just shorthand notation for the projective curve
>      S Y^2 = X^3 + A X S^2 + B S^3,
> where x = X/S, y = Y/S [I am using S instead of the usual Z here,
> since z is about to be used to describe that birational transformation].
> As usual, the point at infinity on E is (X,Y,S) = (0,1,0).
>    The map given near the bottom of page 16 is:
> (x,y) maps to (z,w), where z = -x/y, w = -1/y,
> from y^2 = x^3 + A x + B to the curve:
> E' : w = z^3 + A w^2 z + B w^3, which is shorthand notation for:
>     T^2 W = Z^3 + A W^2 Z + B W^3,
> where w = W/T, z = Z/T.
> Then (X,Y,Z) maps to (z,w,1) = (-x/y, -1/y, 1)  = (-x, -1, y)
>                        = (-X/S, -1, Y/S) = (-X, -S, Y).
> In summary, using projective form, we are mapping the projective curve:
>    E : S Y^2 = X^3 + A X S^2 + B S^3
> to the projective curve:
>    E' : T^2 W = Z^3 + A W^2 Z + B W^3
> and (X,Y,S) maps to (Z,W,T), where Z = -X, W = -S, T = Y.
> So (0,1,0) on E maps to (0,0,1) on E'.
> That is to say (putting it back into affine form), the point at infinity 
> on y^2 = x^3 + A x + B maps to (0,0) on w = z^3 + A w^2 z + B w^3,
> as claimed at the top of page 17.
> 
>    Of course, as an intuitive alternative, you can see it is natural
> that the point at infinity on E should map to (0,0) on E', since
> when x,y go to infinity, -1/y clearly goes to 0, and
> (-x/y)^2 = x^2 / (x^3 + A x + B) goes to 0, and so -x/y goes to 0.
> 
>    - Victor
> 
> On Thu, 13 Apr 2006, Charlotte Griffin wrote:
> > Victor,
> >
> > I'm stuck on one point of the elliptic curves notes, I wonder if you could
> > explain what I'm missing:
> >
> > The very first line on page 17 - that the point at infinity maps to (0,0).
> > I don't see why.
> >
> > I'm sure it's something obvious and I'll kick myself when I find out.
> >
> > Thank you, and Happy Easter,
> >
> > Charlotte
> >
>