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There are genus one curves of every index over every number field
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Pete L Clark
Published/Copyright:
May 17, 2006
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 genus one curves of every index over every number field.
Received: 2004-11-18
Revised: 2005-05-13
Published Online: 2006-05-17
Published in Print: 2006-05-01
© Walter de Gruyter
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Articles in the same Issue
- Traces of CM values of modular functions
- Représentations semi-stables de torsion dans le cas er < p − 1
- Zero cycles on a threefold with isolated singularities
- On the density of algebraic foliations without algebraic invariant sets
- Compactness of solutions to some geometric fourth-order equations
- Relative cohomology with respect to a Lefschetz pencil
- There are genus one curves of every index over every number field
- The uniqueness of Cuntz-Krieger type algebras
