Using elliptic curves to produce quadratic number fields of high three-rank
Several previous authors have constructed infinite families of quadratic fields possessing non-trivial lower bounds on their three-ranks. In this paper the author gives new and shorter proofs of the results on some of these families. The method relies on the known connection between the arithmetic of the elliptic curves of the form \(y^2=x^3+d\) and the arithmetic of the fields \(\mathbb Q(\sqrt d)\) and \(\mathbb Q(\sqrt{-3d})\). The method is illustrated in detail on a family of \textit{D. Shanks} [J. Number Theory 4, No. 6, 537--556 (1972; Zbl 0265.12001)]. Other previously discovered polynomials are treated in the new context, and it is shown how one can produce infinitely many such polynomials. Finally, specific examples of quadratic fields with high three-rank are given.
scientific article; zbMATH DE number 2104653
scientific article