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A formula for the Selmer group of a rational three-isogeny

Matt DeLong
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Abstract

A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y^2=x^3+a(x-b)^2. The formula is in terms of the three-ranks of the quadratic number fields Q(\sqrt{a}) and Q(\sqrt{-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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Matt DeLong. 1999-06-10. A formula for the Selmer group of a rational three-isogeny. https://arxiv.org/abs/math/9906068

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