Estimating the 2-rank of Cubic Fields by Selmer Groups of Elliptic Curves

Abstract

Frey and his coauthors have established a relationship between the 2-torsion of the Selmer group of an elliptic curve of the special formE:y2=x3±k2and the 2-class number of pure cubic fieldK=Q((∓k2)1/3)=Q((∓k)1/3).In the present paper we prove a far-reaching generalization of an analogous relationship between the 2-rank of any non-Galois cubic number fieldKand the 2-torsion of the Selmer group of a corresponding elliptic curve. We implemented the resulting algorithm and used it, e.g., to produce four cubic number fields of exact 2-rank 7. The 2-rank of number fields is of special interest because if it is sufficiently large the number field has an infinite class field tower. In particular, the four fields of 2-rank 7 turn out to have infinite class field towers.