Computing torsion for plane quartics without using height bounds

Raymond van Bommel

Published 2023-01-19Version 1

We describe an algorithm that provably computes the rational torsion subgroup of the Jacobian of a curve without relying on height bounds. Instead, it relies on computing torsion points over small number fields. Both complex analytic and Chinese remainder theorem based methods are used to find such torsion points. The method has been implemented in Magma and used to provably compute the rational torsion subgroup for more than 98% of Jacobians of curves in a dataset due to Sutherland consisting of 82240 plane quartic curves.