Non-existence of points rational over number fields on Shimura curves

TL;DR

This paper extends the non-existence results of rational points on Shimura curves from imaginary quadratic fields to higher degree number fields and provides counterexamples to the Hasse principle.

Contribution

It generalizes previous work to broader number fields and introduces counterexamples to the Hasse principle on Shimura curves.

Findings

Shimura curves have no rational points over higher degree number fields under certain conditions.

Counterexamples to the Hasse principle are constructed for Shimura curves.

Results expand understanding of rational points on Shimura curves beyond imaginary quadratic fields.

Abstract

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we expand their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.