Some simple elliptic surfaces of genus zero (Q1191498)

This paper investigates the elliptic curves \(y^ 2=x^ 3+k\) over \(\mathbb{Q}(u)\). First, for \(k\in\mathbb{C}(u)\) of degree at most three, the author determines a basis for the group of solutions over \(\mathbb{C}(u)\). Then, via Galois theory, the author finds a basis for the group of solutions over \(\mathbb{Q}(u)\) when \(k\in\mathbb{Q}(u)\). Here the rank may be 0, 1, or 2, and the author completely characterizes which polynomials \(k\) correspond to each possible rank. Finally, the author determines the set of all \(\mathbb{Q}[u]\)- integral solutions. There are at most three pairs \((x,\pm y)\) of such integral solutions, and the polynomials \(k\) corresponding to each possibility are again parametrized. The paper gives proofs for the above results which are complete except for some routine computations. These results had been previously announced in \textit{A. Bremner} [Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 2-26 (1989; Zbl 0689.14011)].