## Algebra & Number Theory

### Period, index and potential, III

#### 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 $C∕K$ 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 $P2$, 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.

#### Article information

Source
Algebra Number Theory, Volume 4, Number 2 (2010), 151-174.

Dates
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
https://projecteuclid.org/euclid.ant/1513729506

Digital Object Identifier
doi:10.2140/ant.2010.4.151

Mathematical Reviews number (MathSciNet)
MR2592017

Zentralblatt MATH identifier
1200.11037

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]

#### Citation

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

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