Re: question on heightmatrix for algebraic points

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Jun 25, 2025 at 11:30:45AM +0100, John Cremona wrote:
> 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

Thanks John for the clarification!
Thus the PARI/GP documentation is correct:

?? ellheight(E,{P},{Q}):

* If the argument P \in E(K) is present,  returns the global N'eron-Tate height
* h(P) of the point,  using the normalization in Cremona's
Algorithms for modular elliptic curves.

I added a clarification.

 The canonical height is equal to the N'eron-Tate height divided by the degree
of the number field.   For a curve over a number field, it can be computed with
ellheight(E,P{,Q})/#E.nf.zk.

Cheers,
Bill.