Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

Pete L. Clark, James Stankewicz

Published 2016-12-05Version 1

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 $\mathbb Q$.