Re: question on heightmatrix for algebraic points

[John Cremona <john.cremona@gmail.com>]

It is usual in the formula for the canonical height of points over
number fields to divide by the degree of the extension, the reason
being that the height thus obtained is independent of which field you
consider the point being defined over.  (For example, rational points
then have the same height even if you consider them to be defined over
a number field).

One place where this is not the right thing to do is when you use the
heights to define the regulator which appears in the Birch
Swinnerton-Dyer conjecture.  Then, you should not normalise the
heights.  The non-normalised height is called the "Neron-Tate height".
See https://www.lmfdb.org/knowledge/show/ec.regulator

John

On Tue, 24 Jun 2025 at 21:07, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Tue, Jun 24, 2025 at 12:30:16PM -0700, American Citizen wrote:
> > Hello, all of you, I appreciate your patience, especially Bill's.
> >
> > I have some questions about heights of algebraic points on an elliptic curve
> > and the height matrix associated with them.
> >
> > Let an elliptic curve E be expressed in Weierstrass format.
> >
> > (1)  E = [0, 0, 0, 100, 0]
> >
> > One Mordell-Weil basis for E is the point [5,25].
> >
> > We define 2 algebraic points on E
> >
> >    p = [1, sqrt(101)] with height ~= 4.69969906449875... using
> > K1=nfinit(x^2-101) and ellinit(e,K1)
> >    q = [2, sqrt(208)] with height ~= 2.37364501798303... using
> > K2=nfinit(x^2-208) and ellinit(e,K2)
>
> You need to construct the compositum of your fields, so that you
> have a common field for both.
>  Do this:
>
>  ? [P,a,b]=polcompositum(x^2-101,x^2-208,1)[1];
>  ? a^2
>  %5 = Mod(101,x^4-618*x^2+11449)
>  ? b^2
>  %6 = Mod(208,x^4-618*x^2+11449)
>  ? K=nfinit(P);
>  ? E=ellinit([0, 0, 0, 100, 0],K);
>  ? p=[1,a]; q=[2,b];
>  ? ellheightmatrix(E,[p,q])
>  %7 = [9.3993981289975127877576183833519503910,4.701977403289150032E-38;
>        4.701977403289150032E-38,4.7472900359660764922371320463865657704]
>  ? elladd(E,p,q)
>  %8 = [Mod(x^2 - 3, x^4 - 618*x^2 + 11449), Mod(-213/214*x^3 + 345/214*x, x^4 - 618*x^2 + 11449)]
>
> Cheers,
> Bill
>