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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 $C$?

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    $\begingroup$ In view of the BSD conjecture, $f^{(m)}(1)$ should be related to the regulator of the elliptic curve, i.e., a determinant made 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
  • $\begingroup$ @ACL: There is an error in the question: the constant is $v$ without known relation to rational numbers. $\endgroup$ – China-Hong Kong Jun 5 '14 at 15:38
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    $\begingroup$ Then this looks worse. Set $\nu=f^{(m)}(1)/h(S)$; the question begins: from a rational point of infinite order, construct other points. $\endgroup$ – ACL Jun 5 '14 at 15:43
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As ACL noted, $f^{(m)}(1)$ is supposed to be a multiple of the height regulator, not the height of a single point. Assuming Birch-Swinnerton-Dyer, one can in principle use the value of $f^{(m)}(1)$ to assist in searching for rational points (but not to write down a formula), as explained in

Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights, Math Comp. 68 (1999), 835-858.

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  • $\begingroup$ @ Joe Silverman: There is an error in the question: the constant is $v$ without known relation to rational numbers. $\endgroup$ – China-Hong Kong Jun 5 '14 at 15:40
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    $\begingroup$ @Shpigle I now don't understand what you're asking, unless it's as ACL suggests, namely use the value of $f^{(m)}(1)/h(S)$ to construct other points. As in my paper, if for example rank is 2 and you know one rational point $S$, then you can use $f^{(2)}(1)$ and $S$ to help search for an independent point. $\endgroup$ – Joe Silverman Jun 5 '14 at 19:02

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