Shimura curves and explicit descent obstructions via level structure

Abstract

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the descent (or equivalently \'etale Brauer-Manin) obstruction to rational points applied to \'etale coverings coming from the level structure. More precisely, we find infinitely many quadratic fields defined using congruence conditions such that a twist of a related Shimura curve by each of those fields violates the Hasse principle. As a minimal example, we find the twist of the genus 11 Shimura curve $X^{143}$ by $\mathbf{Q}(\sqrt{-67})$ and its bi-elliptic involution to violate the Hasse principle.