Algebra & Number Theory

Period, index and potential, III

Pete L. Clark and Shahed Sharif

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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.

Article information

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

period index Tate–Shafarevich group


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.

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