One of the techniques used to quantifying the size of a point on an elliptic curve is the so called **canonical height** defined as follow: Let $R=(x,y)∈C(ℚ)$ where $x=(p/d),p,d∈ℤ$. Define the *naive* or *Weil height* of $R$ as

$$h(R)=log(max{|p|,|d|})$$

Then the canonical height $h$ is defined uniquely as map from $C(ℚ)$ to $ℝ$ by

$$h(R)=lim_{n→∞}((h(2ⁿR))/4ⁿ)$$

Gross and Zagier (1983, 1986) proved that if $C$ is a modular elliptic curve over $ℚ$, and $P$ is a the **Heegner point** then $$f′(1)=c.h(P),c≠0$$ Thus $P$ has infinite order if and only if $f′(1)≠0$. More generally, Gross, Kohnen & Zagier (1987) showed that Heegner points could be used to **construct rational points** on the curve for each positive integer $n$, and the heights of these points were the coefficients of a modular form of weight $3/2$.

Assume that (this is just an assumption and it is never proved) the analytic rank $m≥2$ verify this equation: $$f^{(m)}(1)=v.h(S)$$ where $S$ is a rational point of infinite order and $v≠0$ is a real number. My **question** is: Can we use this formula to construct rational points on the curve $C$?

determinantmade from heights of points out of a basis of the Mordell-Weil group (mod torsion). So it looks unlikely that one could prove that $f^{(m)}(1)$ is a rational multiple of the height of a single point. $\endgroup$ – ACL Jun 5 '14 at 10:29