Three constructions of rational points on \(Y^ 2=X^ 3\pm NX\) (Q1188015)

Mordell conjectured in 1967 that the elliptic curve \(Y^ 2=X^ 3+pX\) has a rational point of infinite order provided \(p\) is a prime \(\equiv 5(8)\). This paper establishes the conjecture for \(p\equiv 5(16)\). More generally suppose \(p\) is prime and \(N=p\) or \(p^ 3\). It is shown that \(Y^ 2=X^ 3+NX\) has rank 1 if \(N\equiv 3\) or 5(16) and that \(Y^ 2=X^ 3-NX\) has rank 1 if \(N\equiv 5\) or 7(16). The desired rational point is constructed à la Heegner; by evaluating Weber's modular functions of levels 8 and 16 at \(ip\) one gets a real point on the desired curve, rational over an abelian extension of \(Q(i)\), and the trace of this point to \(Q(i)\) is shown to have infinite order. A refinement of results of Weber and Birch on the values of Weber's functions at \(ip\), established by means of Shimura reciprocity, plays a central part in the proof. Like Satgé's work on the Sylvester problem and the author's earlier result on congruent numbers, this paper suggests the existence of a more encompassing theory of \(CM\) points, going beyond the Heegner points of Birch, Gross and Zagier.