computing height on elliptic curve of the form $y^2=x^3-nx$

Question

Let $E$ be the elliptic curve $$y^2 =x^3 - 19*67 x$$ and $P=[26011/625,2159616/15625]$, I want to compute $\hat{h}(P)$ using formula given in

Fujita, Y., & Terai, N. (2011). Generators for the elliptic curve $ y^ 2= x^ 3-nx$. Journal de théorie des nombres de Bordeaux, 23(2), 403-416.

To compute Archimedean part (formula (3.2) in the article), we should to compute $\log{\theta}$, there is a formula in the article: $$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)/2}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))}$$ Is there a command in Pari/GP or Magma to compute this value? As mentioned in the article, $|\theta|<1/(1-q)$, is there a lower bound for $|\theta|$ in elliptic curves of the form $y^2=x^3-nx$?

Answer 1

Is there some reason that you want to use this formula? You can get the height directly in PARI with the commands:

gp > EE=ellinit([0,0,0,-19*67,0]);

gp > PP=[26011/625,2159616/15625];

gp > ellheight(EE,PP)

%1 = 8.6541981

For the local archimedean height, the following formula is in Advanced Topics in the Arithmetic of Elliptic Curves Theorem VI.3.4. This uses the product form of the Weierstrass sigma function (theta function), rather than the series that you're using, but it still converges quite rapidly.

Theorem Let $u=e^{2\pi i z}$ and $q=e^{2\pi i\tau}$. Then $$ \lambda(z)=-\frac12B_2\left(\frac{\operatorname{Im} z}{\operatorname{Im}\tau}\right) \log|q|-\log|1-u|-\sum_{n\ge1}\log|(1-q^nu)(1-q^nu^{-1})|, $$ where $B_2(T)=T^2-T+\frac16$.

Regarding your question about a lower bound for your $\theta$, I think that the answer is no. In your notation, you have $\lambda(z)=-\log|\theta(z)|$, so your upper bound for $|\theta(z)|$ gives a lower bound for the height. But the height has a logarithmic pole at the identity element, and if $E(\mathbb{Q})$ has rank at least one, then the rational points are dense in the (identity component of the) real points. So in your case there are multiples of your point that are arbitrarily close to the identity (in the real topology), hence their archimedean heights can be arbitrarily large.

Using results from linear forms in logs, one can get some sort of upper bound for $\lambda(mz)$ that depends on $m$ and $z$ and $\tau$, but the limsup of this bound over $m$ is $\infty$.