Vol. 4, No. 2, 2010

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Period, index and potential, III

Pete L. Clark and Shahed Sharif

Vol. 4 (2010), No. 2, 151–174
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 $\left(P,I\right)$ 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
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