Two-Torsion Subgroups of some Modular Jacobians

TL;DR

This paper presents a practical computational method for determining the 2-torsion subgroup of Jacobians of certain algebraic curves, leveraging theta hyperplanes and numerical techniques, and applies it to verify a conjecture for specific modular Jacobians.

Contribution

The paper introduces a new explicit method to compute 2-torsion subgroups of Jacobians of non-hyperelliptic curves of genus 3 to 5, using theta hyperplanes and numerical approximations.

Findings

Successfully computed 2-torsion subgroups for specific modular Jacobians.

Verified the generalized Ogg conjecture for N=42, 55, 63, 72, 75.

Demonstrated the practicality of the method with explicit examples.

Abstract

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the $2$-torsion subgroup. Using $p$-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton-Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the $2$-torsion points. We demonstrate the practicality of our method by computing the $2$-torsion of the modular Jacobians $J_{0}\left( N \right)$ for $N = 42, 55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg conjecture for these values.