Computing modular equations for Shimura curves

TL;DR

This paper introduces a method to compute modular equations for Shimura curves, extending classical modular relations to a broader setting using explicit Hecke operator computations.

Contribution

It develops an explicit method for computing modular equations on Shimura curves, generalizing classical modular polynomial computations.

Findings

Provides a new computational approach for Shimura curve modular equations

Utilizes explicit Hecke operator calculations on Shimura modular forms

Enables practical computation of modular relations in the Shimura setting

Abstract

In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(\tau)$ and $j(N\tau)$, where $j(\tau)$ is the standard elliptic $j$-function. In this paper, we will describe a method to compute modular equations in the setting of Shimura curves. The main ingredient is the explicit method for computing Hecke operators on the spaces of modular forms on Shimura curves developed in [13].