  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.
