Algebra & Number Theory
- Algebra Number Theory
- Volume 4, Number 2 (2010), 151-174.
Period, index and potential, III
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 such that is divisible by and divides , there exists a number field and a genus-one curve with period and index . Second, let be any elliptic curve over a global field , and let be any integer indivisible by the characteristic of . We construct infinitely many genus-one curves with period , index , and Jacobian . 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.
Algebra Number Theory, Volume 4, Number 2 (2010), 151-174.
Received: 15 January 2009
Revised: 12 November 2009
Accepted: 16 November 2009
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Clark, Pete L.; Sharif, Shahed. Period, index and potential, III. Algebra Number Theory 4 (2010), no. 2, 151--174. doi:10.2140/ant.2010.4.151. https://projecteuclid.org/euclid.ant/1513729506