Equations of hyperelliptic Shimura curves

TL;DR

This paper determines explicit equations for hyperelliptic Shimura curves using Borcherds forms and Schofer's formula, addressing both algebraic and modular form realization problems.

Contribution

It introduces a method to explicitly compute equations of hyperelliptic Shimura curves via Borcherds forms and solves related modular form realization questions.

Findings

Explicit equations for all hyperelliptic Shimura curves $X_0^D(N)$ are obtained.

A new approach using integer programming to construct Borcherds forms is developed.

The paper addresses the realization of certain modular forms as Borcherds forms on Shimura curves.

Abstract

By constructing suitable Borcherds forms on Shimura curves and using Schofer's formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_0^D(N)/W_{D,N}$ with a divisor supported on CM-divisors can be realized as a Borcherds form, where $X_0^D(N)/W_{D,N}$ denotes the quotient of $X_0^D(N)$ by all the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.