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 étale Brauer-Manin) obstruction to rational points applied to étale 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.

The minimal example is incorrect. There is an error in the discussion about etale covers of twists, but this error has informed further progress, to appear in a new manuscript