Vol. 4, No. 2, 2010

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12, 1 issue

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Period, index and potential, III

Pete L. Clark and Shahed Sharif

Vol. 4 (2010), No. 2, 151–174

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.

period, index, Tate–Shafarevich group
Mathematical Subject Classification 2000
Primary: 11G05
Received: 15 January 2009
Revised: 12 November 2009
Accepted: 16 November 2009
Published: 26 January 2010
Pete L. Clark
University of Georgia
Department of Mathematics
Athens, GA 30602
United States
Shahed Sharif
Department of Mathematics
Duke University
Durham, NC 27708
United States