A formula for the Selmer group of a rational three-isogeny
Abstract
A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y2=x3+a(x-b)2. The formula is in terms of the three-ranks of the quadratic number fields Q(a) and Q(-3a) and various aspects of the arithmetic of these number fields. In addition 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.
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