Computations on Modular Jacobian Surfaces

TL;DR

This paper develops a method to find rational equations of genus 2 curves with Jacobians linked to abelian varieties associated with modular forms, and catalogs all such curves for levels up to 500.

Contribution

It introduces a new computational approach to explicitly determine genus 2 curves with specific Jacobian structures related to modular forms.

Findings

All such curves for N ≤ 500 are explicitly listed.

The method effectively finds rational equations for these genus 2 curves.

Provides a comprehensive catalog of genus 2 curves with particular Jacobian properties.

Abstract

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties $A_f$ attached by Shimura to normalized newforms $f \in S_2( \Gamma_0(N))$. We present all the curves corresponding to principally polarized surfaces $A_f$ for $N\leq500$.