Curves over every global field violating the local-global principle
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There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Mathematical Sciences
Poonen, B. “Curves over Every Global Field Violating the Local-global Principle.” Journal of Mathematical Sciences 171.6 (2010): 782–785. Web. 27 Apr. 2012. © 2010 Springer Science+Business Media B.V.
Author's final manuscript