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Local-global principles for Weil–Châtelet divisibility in positive characteristic

Published online by Cambridge University Press:  01 February 2017

BRENDAN CREUTZ  and
JOSÉ FELIPE VOLOCH
BRENDAN CREUTZ
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand. e-mail: brendan.creutz@canterbury.ac.nz
JOSÉ FELIPE VOLOCH
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand and Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A. e-mail: felipe.voloch@canterbury.ac.nz
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Abstract

We extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 357 - 367
Copyright
Copyright © Cambridge Philosophical Society 2017 

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