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There are genus one curves of every index over every infinite, finitely generated field

  • Pete L. Clark EMAIL logo and Allan Lacy
Published/Copyright: September 20, 2016

Abstract

We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E/K which admits infinitely many torsors with index any given n>1.

Acknowledgements

This work had its genesis in a conversation with Shahed Sharif. In particular, the main idea of the proof of Lemma 7.4 is due to him. We thank Cristian D. González-Avilés for technical (and moral) support. We are grateful to Kestutis Česnavičius for help strengthening the statement and clarifying the proof of Theorem 5.2. We thank the referee for a careful reading.

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Received: 2014-08-20
Revised: 2015-05-28
Published Online: 2016-09-20
Published in Print: 2019-04-01

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