John Cremona on Fri, 11 Jan 2002 10:35:18 +0000 |
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Re: Mordell Weil generator for y^2=x^3+7823 |
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 -- Prof. J. E. Cremona | University of Nottingham | Tel.: +44-115-9514920 School of Mathematical Sciences | Fax: +44-115-9514951 University Park | Email: John.Cremona@nottingham.ac.uk Nottingham NG7 2RD, UK |