Estimating the 2-rank of cubic fields by Selmer groups of elliptic curves

[摘要] Frey and his coauthors have established a relationship between the 2-torsion of the Selmer group of an elliptic curve of the special form E: y(2) = x(3) +/- k(2) and the 2-class number of pure cubic field K = Q((-/+k(2))(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 field K and 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. (C) 1997 Academic Press.

[发布日期] 1997-02-01