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

Pete L. Clark

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

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arxiv math/0411413 v1 pith:OUI7AZUN submitted 2004-11-18 math.NT math.AG

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

Pete L. Clark This is my paper

classification math.NT math.AG

keywords everycurvesindextherefieldgenusnumberrational

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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.

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

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