There are genus one curves of every index over every number field

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We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.
5 pages

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Number Theory, Algebraic Geometry

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http://salesiana.dossiersoluciones.com/handle/123456789/73286

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0411413
 http://arxiv.org/abs/math/0411413

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arXiv