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arXiv · 0811.3019

Period, index and potential sha

Pete L. Clark·Shahed Sharif
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Abstract

In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E 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 over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension.

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Pete L. Clark, Shahed Sharif. 2008-11-18. Period, index and potential sha. https://arxiv.org/abs/0811.3019

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