The selmer groups of elliptic curves and the ideal class groups of quadratic fields

Abstract

Let D be an integer. Consider the elliptic curve E/Q :y² = x³ + D, which has j-invariant 0. We can show that for this elliptic curve the rank of its 3-Selmer group is closely related to the 3-rank of the ideal class groups of the quadratic fields and . Fol the same family of curves Frey showed that, if D is a cube, the rank of the Selrner group of a 3-isogeny is related to the class number of the quadratic field [3]. Also Jan Nekevá[rbreve] proved some analogous result for elliptic curve given by Dy² = 4x³ − 27 which is isomorphic to the curve given by y² = x³ − 432D³ [4]. Our method is different from theirs and it can give a far more complete result for general D.