Re: Mordell Weil generator for y^2=x^3+7823

[John Cremona <John.Cremona@nottingham.ac.uk>]

I agree with Michael.  One piece of evidence for his last comment

> In terms of measuring computational difficulty, I think the x-coordinate
> height is more appropriate, since it measures the size of the actual
> coordinates we are looking for.

is clear if you look at the bound between the canonical height and the
"naive" or Weil height, which for a point P=(x,y) is
h(P)=log(max(num(x),den(x))).  With the larger normalization for h^(P),
which I use, the quantity which is bounded (for a given curve) is
|h(P)-h^(P)|, so h^(P) is telling you about the naive size of the point
(up to the bounded error).  With the smaller normalization it is
|h(P)-(1/2)h^(P)| which is bounded (see Silverman's papers on this
bound, for example).

Years ago I suggested to Henri Cohen that he change the normalization
used in Pari, as I got tired of having to type 2*hell(e,p) (as the
function was then called).  He did not change hell (noe ellheight) at
all, but did change the height pairing, so that one can compute
regulators in a way compatible with B&SD.  So my gpalias file has the
entry 

ellht(e,p)=2*ellheight(e,p)

Magma uses "my" normalization, since it borrowed my code;  I hope that
David does not want to change that now, as it would break quite a few
programs!

C'est la vie.

John

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