  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 violates the Hasse principle, i.e., has points everywhere
locally but not globally. We show that our conjecture holds for all but
finitely many Shimura curves of the form X^D_0(N)_{/Q} or X^D_1(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.
