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On some elliptic curves with large sha. (English) Zbl 0983.11032
The class of elliptic curves considered in this paper is given by \[ C(n): y^2=x^3+nx, \] where \(n\in \mathbb{Z}\). Let \(p\) be a prime satisfying \(p\equiv 1\pmod 8\). The author is mainly concerned with the case \(C(p^3)\), and also with \(C(p)\), but only because of its connection with the former. These curves have Mordell-Weil rank 0 or 2, provided the Birch and Swinnerton-Dyer conjecture holds (this conjecture is assumed to be true throughout the paper). What the author really is interested in is the order of the Tate-Shafarevich group of \(C(p^3)\), \(\text{sha}(p)\) for short. It turns out that, regularly, the curves \(C(p^3)\) have large sha and this value is relatively often an integer square. For each \(k\leq 105\) the author calculates the primes \(p< 10^5\) for which \(\text{sha}(p)= k^2\) (all values of \(k\) occur). The largest sha found by the author is \(\text{sha} (4105033)= 635^2\).
11G05 Elliptic curves over global fields
14H52 Elliptic curves
11D25 Cubic and quartic Diophantine equations
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