Relating elliptic curves to three-ranks of quadratic number fields.

Abstract

A formula is given for the dimension of the Selmer group of the non-constant j-invariant case of a rational three-isogeny of elliptic curves, in terms of the three-ranks of two associated quadratic number fields and various aspects of the arithmetic of these number fields. A duality theorem is used to relate the dimension of the Selmer group of the three-isogeny with the dimension of the Selmer group of its dual isogeny. Similar results of Satge are extended for the special case when the j-invariant of the elliptic curve is 0. These results are used to translate some of the machinery of rational elliptic surfaces to obtain old and new results on polynomials which give rise to infinite families of quadratic fields with non-trivial three-ranks. Infinitely many polynomials, each giving rise to infinitely many imaginary quadratic fields with three-rank at least two, are produced.