Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves

TL;DR

This paper develops a method to analyze rational points on quotients of Shimura curves by Atkin-Lehner involutions, providing explicit conditions for triviality and demonstrating an infinite family satisfying these conditions.

Contribution

It introduces a new approach for studying rational points on Shimura curve quotients and identifies explicit conditions for triviality, including an infinite family of examples.

Findings

Explicit conditions for trivial rational points

An infinite family of Shimura curve quotients with trivial rational points

Method applicable to broader classes of Shimura curves

Abstract

In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin-Lehner involutions. We obtain explicit conditions on such quotients for rational points to be ``trivial'' (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions.