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2010 Period, index and potential, III
Pete L. Clark, Shahed Sharif
Algebra Number Theory 4(2): 151-174 (2010). DOI: 10.2140/ant.2010.4.151

Abstract

We present three results on the period-index problem for genus-one curves over global fields. Our first result implies that for every pair of positive integers (P,I) such that I is divisible by P and divides P2, there exists a number field K and a genus-one curve CK with period P and index I. Second, let EK be any elliptic curve over a global field K, and let P>1 be any integer indivisible by the characteristic of K. We construct infinitely many genus-one curves CK with period P, index P2, and Jacobian E. Our third result, on the structure of Shafarevich–Tate groups under field extension, follows as a corollary. Our main tools are Lichtenbaum–Tate duality and the functorial properties of O’Neil’s period-index obstruction map under change of period.

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Pete L. Clark. Shahed Sharif. "Period, index and potential, III." Algebra Number Theory 4 (2) 151 - 174, 2010. https://doi.org/10.2140/ant.2010.4.151

Information

Received: 15 January 2009; Revised: 12 November 2009; Accepted: 16 November 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1200.11037
MathSciNet: MR2592017
Digital Object Identifier: 10.2140/ant.2010.4.151

Subjects:
Primary: 11G05

Keywords: Index , period , Tate–Shafarevich group

Rights: Copyright © 2010 Mathematical Sciences Publishers

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Vol.4 • No. 2 • 2010
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