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 $\mathbb {Q}$. More precisely, the number of squarefree $d$ with $|d| \leq X$ such that the quadratic twist of $(X^D,w_D)$ by $\mathbb {Q}(\sqrt {d})$ violates the Hasse Principle is $\gg$ $X/\log ^{\alpha _D} X$ and $\ll X/\log ^{\beta _D} X$ for explicitly given $0 < \beta _D < \alpha _D < 1$ such that $\alpha _D - \beta _D \rightarrow 0$ as $D \rightarrow \infty$.