Michael Stoll on Fri, 11 Jan 2002 11:23:04 +0100

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

Igor Schein asked (on the PARI developers' list):

> On Thu, Jan 10, 2002 at 04:05:07PM -0500, Michael Stoll wrote:
> > Dear NMBRTHRYsts,
> >
> > may I proudly present the generator of the Mordell-Weil group of
> > y^2 = x^3 + 7823 :
> >
> > x = 2263582143321421502100209233517777/11981673410095561^2
> > y =
> > -186398152584623305624837551485596770028144776655756/11981673410095561^3
> >
> > This point has canonical height 77.617773768638... as expected.
>
> ? x=2263582143321421502100209233517777/11981673410095561^2;
> ? y =
> -186398152584623305624837551485596770028144776655756/11981673410095561^3; ?
> ellheight(ellinit([0,0,0,0,7823]),[x,y]) 38.80888688431904198640563870
> ? %*2
> 77.61777376863808397281127740
>
> Where's the factor of 2 coming from?

> I believe that the factor of 2 is accounted for by the difference
> between the height on E and the height of its image in P^1 = E/{\pm 1},
> under the degree 2 function x.  If the function y where used, then
> the height would be scaled by 3.  I think Cremona makes a note of
> this discrepancy in his book.  I consider Silverman's normalization
> (and of Pari) to be the more mathematically correct and intrinsic
> one than that of Cremona (and Magma).
>
> Incidentally there is another factor of two which can arise due
> to the choice of height pairing <P,Q> = h(P+Q) - h(P) - h(Q) or
> <P,Q> = (h(P+Q) - h(P) - h(Q))/2, and the regulator of n points
> differs by a factor of 2^n depending on this choice, and one must
> take the right definition for the BSD conjecture to hold.  It is
> my understanding however that the factor of 2 has its origin in
> the degree of the x function rather than the definition of the
> height pairing.

> There are two normalizations of heights in use.  The one which gives
> 77.6 is one;  Pari uses the other for heights, though makes the
> adjustment for the height pairing (so as to give appropriate values for
> Birch-Swinnerton-Dyer applications).
>
> Put another way, although Pari's height for this point is 77.6/2=38.8,
> its value for the height pairing is 77.6.

This is correct. As David points out, the mathematically better definition
is the one associated to the divisor class defining the principal
polarisation, which is the class of a point, rather than twice a point,
which gives the x-coordinate map. On the other hand, as John points out,
you then have to define the height pairing as <P,Q> = h(P+Q) - h(P) - h(Q)
in order to make the Birch and Swinnerton-Dyer conjecture work (numerically),
so that <P, P> = 2 h(P), which is the x-coordinate height again.

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.

Michael Stoll