Math @ Duke

Publications [#167240] of Shahed Sharif
Papers Accepted
 P. Clark and S. Sharif, Period, index, and potential Sha,
Algebra and Number Theory
(Fall, 2009), ISSN 19370652 [pdf]
(last updated on 2009/12/17)
Abstract: We present three results on the periodindex problem for genus one curves over global fields. Our first result implies that for every pair of positive integers $(P,I)$ with $P  I  P^2$, 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 $P^2$, and Jacobian $E$. Our third result, on the structure of ShafarevichTate groups under field extension, follows as a corollary. Our main tools are LichtenbaumTate duality and the functorial properties
of O'Neil's periodindex obstruction map under change of period.


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