Computing torsion for plane quartics without using height bounds

Abstract

We describe an algorithm that provably computes the rational torsion subgroup of the Jacobian of a curve without relying on height bounds. Instead, the strategy is to find upper bounds for the torsion subgroup using reduction modulo primes, and searching for torsion points, not just over Q but also over small number fields, until the two bounds meet. Both complex analytic and Chinese remainder theorem based methods are used to find such torsion points. The method has been implemented in Magma for plane quartic curves over Q with a rational point and used to provably compute the rational torsion subgroup for more than 98% of Jacobians of curves in a data set due to Sutherland consisting of 82240 plane quartic curves.