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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

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