Two-Torsion Subgroups of some Modular Jacobians

TL;DR

The work develops and implements a practical framework to compute the full 2-torsion subgroup of Jacobians of non-hyperelliptic curves of genus 3–5 by exploiting the theta-hyperplane correspondence. It constructs zero-dimensional schemes of theta hyperplanes (bitangents, tritangents, quadritangents) and recovers exact 2-torsion points via high-precision approximations and lattice reductions, both in $p$-adic and complex settings. The method is applied to modular Jacobians $J_0(N)$ for $N=42,55,63,72,75$, enabling explicit 2-torsion data and enabling verification of the generalized Ogg conjecture for these values (with 55 treated partially). The results demonstrate the practicality of theta-hyperplane techniques for concrete arithmetic of modular Jacobians and provide a computational blueprint for similar problems in higher genus.

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.

Theorems & Definitions (14)

• Theorem 1
• Definition
• Theorem 2
• Definition
• Theorem 3
• Remark 1
• Remark 2
• Theorem 4
• Remark 3
• Remark 4