Geodesic
Curves over every global field violating the local-global principle
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 141-147 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\#X(k)=n$. Bibl. 26 titles. 
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B. Poonen. Curves over every global field violating the local-global principle. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 141-147. http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a13/

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