Equations of hyperelliptic Shimura curves

TL;DR

This paper presents an algorithm for computing explicit models of hyperelliptic Shimura curves and their quotients, utilizing advanced uniformization and endomorphism formulas, and provides verified equations for these curves.

Contribution

It introduces a novel algorithm that combines non-archimedean uniformization, Gross-Zagier formulas, and Ribet's bimodules to explicitly compute hyperelliptic Shimura curves.

Findings

Algorithm successfully computes explicit equations for Shimura curves.

Provides a verified list of equations matching previous conjectures.

Enhances understanding of the structure of hyperelliptic Shimura curves.

Abstract

We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefnite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our algorithm that had been conjectured by Kurihara.