Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the infinite point. We assume that $P$ is not torsion.


The point $P$ can be written as $(\frac{x}{d^2}, \frac{y}{d^3})$ for some integers $x, y, d$ such that $d$ is prime to $x$ and $y$.


Now consider the sequence of points $P, 2P, 3P, \cdots$, we are led to three sequences $x\_n, y\_n, d\_n$, such that $nP = (\frac{x\_n}{d\_n^2}, \frac{y\_n}{d\_n^3})$.


The question: does the limit $\lim\_{n\rightarrow\infty} \frac{2\log(d\_n)}{n^2}$ exist? Is it equal to the canonical height of $P$?


Same question for $x\_n$ and $y\_n$?




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A related question:


Let us denote by $h(P)$ the natural height of $P$, i.e. $h(P) = \log(\max\{|x|, |d^2|\})$. Does the limit $\lim\_{n\rightarrow\infty}\frac{h(nP)}{n^2}$ exist? (If it exists, then it of course is equal to the canonical height.)



Heights of multiples of rational points on elliptic curves
You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of *Arithmetic of Elliptic Curves*, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since it's stated for number fields, but in your notation, one has
$$
\lim\_{n\to\infty} \frac{2\log d\_n}{n^2}
= \lim\_{n\to\infty} \frac{\log |x\_n|}{n^2}
= \hat h(P).
$$
Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition):
$$ \lim\_{n\to\infty} \frac{\log|x\_n|}{\log d\_n^2} = 1. $$


BTW, the sequence $(d\_n)\_{n\ge1}$ is called the *Elliptic Divisibility Sequence* associated to the curve $E$ and point $P$. The fact that $\log d\_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d\_n$ and $p\nmid d\_m$ for all $m < n$.


