Hasse principle violations for atkin-lehner twists of shimura curves

Let D > 546 be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields l/ℚ such that the twist of the Shimura curve X^D by the main Atkin-Lehner involution w_Dand l/ℚ violates the Hasse Principle over ℚ. More precisely, the number of squarefree d with |d| ≤ X such that the quadratic twist of (X^D,_wD) by ℚ(√d) violates the Hasse Principle is ≫ X/ log^αDX and ≪ X/ log^βDX for explicitly given 0 < β_D< α_D< 1 such that α_D− β_D→ 0 as D→∞.