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 (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.

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