Two-torsion subgroups of some modular Jacobians

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)(N) for N = 42, 55, 63, 72, 75. As a result of this we are able to verify the generalized Ogg conjecture for these values.