There are genus one curves of every index over every number field
Number Theory
2007-05-23 v1 Algebraic Geometry
Abstract
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.
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Cite
@article{arxiv.math/0411413,
title = {There are genus one curves of every index over every number field},
author = {Pete L. Clark},
journal= {arXiv preprint arXiv:math/0411413},
year = {2007}
}
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5 pages