Two-Torsion Subgroups of some Modular Jacobians

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.

corrected proof of theorem 1

References in corpus (1)

• Two-cover descent on hyperelliptic curves

Cited by in corpus (2)

• Three-Torsion Subgroups and Wild Conductors of Genus 3 Hyperelliptic Curves
• Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics