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On the equation $$Y^ 2=X(X^ 2+p)$$. (English) Zbl 0689.14010
Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 3-22 (1989).
[For the entire collection see Zbl 0676.00005.]
Suppose p is a prime number of the form $$8k+5$$. Let $$E_ p$$ denote the elliptic curve given by $$y^ 2=x^ 3+px$$. It is well known that the group of $${\mathbb{Q}}$$-rational points $$E_ p({\mathbb{Q}})$$ is isomorphic to either $${\mathbb{Z}}/2{\mathbb{Z}}$$ or $${\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$$. Moreover, the sign occurring in the functional equation of $$L(E_ p/{\mathbb{Q}},s)$$ is -1. Hence the Birch and Swinnerton-Dyer conjecture predicts that we should have $$E_ p({\mathbb{Q}})\cong {\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$$. (Recently, this has been checked in case $$p\equiv 5$$ mod 16 by Paul Monsky.) - Since the curve $$E_ p$$ has complex multiplication, it is parametrized by a modular curve (in fact $$X_ 0(64p^ 2))$$. This allows one to invoke a theorem of Gross and Zagier, which implies that whenever at $$s=1$$, the derivative $$L'(E_ p/{\mathbb{Q}},1)\neq 0$$, it follows that $$E_ p({\mathbb{Q}})\cong {\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}.$$
The author evaluates this derivative numerically for primes $$p\equiv 5$$ mod 8, $$5\leq p\leq 20,000$$. This allows him to verify the prediction in this range. Moreover, he tabulates the height of a generator of infinite order, again using the result of Gross and Zagier. The case $$p=2437$$ turns out to give a fairly large height (some even larger ones are also found, for bigger primes), and using a descent, the author computes an actual generator in that case. Its x-coordinate involves a numerator and denominator of 64 and 63 digits, respectively.
Reviewer: J.Top

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11D25 Cubic and quartic Diophantine equations 14H52 Elliptic curves