Abstract
Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X D0 (N)/ℚ or X D1 (N)/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.
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Clark, P.L. On the Hasse principle for Shimura curves. Isr. J. Math. 171, 349–365 (2009). https://doi.org/10.1007/s11856-009-0053-6
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DOI: https://doi.org/10.1007/s11856-009-0053-6