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 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.
Citation
Pete L. Clark. Shahed Sharif. "Period, index and potential, III." Algebra Number Theory 4 (2) 151 - 174, 2010. https://doi.org/10.2140/ant.2010.4.151
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