Rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves

TL;DR

This paper computes rational points on Atkin-Lehner quotients of geometrically hyperelliptic Shimura curves, identifying CM points and applying various techniques to analyze these complex algebraic structures.

Contribution

It provides explicit computations of rational points on these quotients and identifies CM points, advancing understanding of their arithmetic properties.

Findings

Rational points on Atkin-Lehner quotients are explicitly determined.

Many rational points are identified as CM points.

The paper applies diverse techniques to analyze these curves.

Abstract

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $\mathbb{Q}$-rational points on the Atkin-Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.