Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

Abstract

Let $D>546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $l/\mathbb{Q}$ such that the twist of the Shimura curve $X^D$ by the main Atkin-Lehner involution $w_D$ and $l/\mathbb{Q}$ violates the Hasse Principle over $\mathbb{Q}$.

11 pages, submitted