Curves over global fields violating the Hasse Principle

Details Curves over global fields violating the Hasse Principle Curves over global fields violating the Hasse Principle

2009-05-21

arxiv.org/abs/0905.3459v1

Mathematics

Number Theory

16 pages; comments welcome

Scientific paper

In response to a question of B. Poonen, we exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. In fact we can find such examples among Atkin-Lehner twists of certain elliptic modular curves and -- in positive characteristic -- Drinfeld modular curves. Our main tool is a refinement of the "Twist Anti-Hasse Principle" (TAHP). We also use TAHP to construct further Hasse Principle violations, for instance among curves over any number field of any given genus g which is at least 2.

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