Un analogue du calcul de Heegner. (An analogue of the Heegner calculus) (Q1821163)

In this interesting paper, the author extends to cubic twists of an elliptic curve Heegner's modular construction of a point of infinite order on quadratic twists. He shows in particular that if p is an odd prime congruent to 2 (resp. 5) mod 9, then, the elliptic curve \(X^ 3+Y^ 3=2pZ^ 3\) (resp. 2p\({}^ 2Z^ 3)\) has infinitely many rational points. The main point in the proof is the construction of a point on an associated curve of genus \(1,\) whose field of definition has degree prime to 3, and which is obtained by evaluating certain modular forms at quadratic integers with large conductors. The end of the proof follows Selmer's descent argument [see \textit{J. W. S. Cassels}, J. Lond. Math. Soc. 41, 193-291 (1966; Zbl 0138.270)].