Computations on Modular Jacobian Surfaces

TL;DR

This paper studies explicit realizations of modular abelian surfaces $A_f$ attached to newforms as Jacobians of genus $2$ curves over $\mathbb{Q}$ for $N\le 500$. It introduces an arithmetic approach based on odd Thetanullwerte to recover hyperelliptic models $C_F: Y^2=F(X)$ with $F\in \mathbb{Q}[X]$, such that $\mathrm{Jac}\,C_F \simeq A_f$ (up to sign) and uses the Eichler–Shimura congruence to fix the sign. The implementation in Magma then yields all modular Jacobian surfaces in the range, producing many minimal integral equations over $\mathbb{Z}[1/2]$ and validating them via Igusa invariants and $L$-series data. The results are tabulated, providing a comprehensive catalog of genus $2$ curves whose Jacobians are isogenous to the two-dimensional factors of $J_0(N)^{\mathrm{new}}$ for $N\le 500$, with potential cryptographic relevance for constructing hyperelliptic curves with controlled arithmetic.

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 $f \in S_2( Γ_0(N))$. We present all the curves corresponding to principally polarized surfaces $A_f$ for $N\leq500$.

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Theorem 1
• Proposition 3