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arXiv:math/0411413 [math.NT]AbstractReferencesReviewsResources

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

Pete L. Clark

Published 2004-11-18Version 1

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.

Comments: 5 pages
Categories: math.NT, math.AG
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