Shimura curves and explicit descent obstructions via level structure

TL;DR

This paper constructs families of Shimura curves with specific level structures that exhibit failures of the Hasse principle due to descent obstructions, providing explicit counterexamples over quadratic fields.

Contribution

It introduces new families of Shimura curves with level structures that produce explicit, infinite counterexamples to the Hasse principle via descent obstructions.

Findings

Infinite families of Shimura curves with no p-adic points for some p.

Explicit counterexamples to the Hasse principle over quadratic fields.

A minimal example involving a genus 11 Shimura curve violating the Hasse principle.

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.