Period, index and potential, III

Clark, Pete L.; Sharif, Shahed

doi:10.2140/ant.2010.4.151

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Period, index and potential, III

Pete L. Clark and Shahed Sharif

Vol. 4 (2010), No. 2, 151–174

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 $P^{2}$ , there exists a number field $K$ and a genus-one curve $C_{∕ K}$ with period $P$ and index $I$ . Second, let $E_{∕ K}$ 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 $C_{∕ K}$ with period $P$ , index $P^{2}$ , 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.

Keywords

period, index, Tate–Shafarevich group

Mathematical Subject Classification 2000

Primary: 11G05

Milestones

Received: 15 January 2009

Revised: 12 November 2009

Accepted: 16 November 2009

Published: 26 January 2010

Authors

Pete L. Clark
University of Georgia Department of Mathematics Athens, GA 30602 United States
http://www.math.uga.edu/~pete/
Shahed Sharif
Department of Mathematics Duke University Durham, NC 27708 United States
http://www.math.duke.edu/~sharif

Vol. 4, No. 2, 2010