Selmer groups for elliptic curves with isogenies of prime degree

Abstract

The Mordell-Weil theorem states that the points of an elliptic curve defined over a number field form a finitely generated, abelian group. The rank of this group, generally referred to as the rank of the elliptic curve, is hard to study. The Selmer group, defined via Galois cohomology, gives a way of approximating the rank of an elliptic curve. The Selmer group is, itself, difficult to study in general.We examine the Selmer group for an elliptic curve which admits an isogeny degree p, for an odd prime p. Using the kernel of the isogeny, and the kernel of its dual isogeny, we give upper and lower bounds on the p-rank of the Selmer group in terms of the arithmetic of certain number fields. We show, by way of examples, that these bounds can be computed for families of quadratic twists of an elliptic curve.For elliptic curves defined over the rational numbers; we examine the relationship between these bounds on the p-rank of the Selmer group and the algebraic Iwasawa invariants associated to the elliptic curve for the cyclotomic Zp-extension of Q.