Computing torsion for plane quartics without using height bounds

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.Comment: Copy of Magma code and data file repository included as ancillary file. Comments always welcome

Efficient computation of BSD invariants in genus 2

Recently, all Birch and Swinnerton-Dyer invariants, except for the order of Sha, have been computed for all curves of genus 2 contained in the L-functions and Modular Forms Database. This report explains the improvements made to the implementation of the algorithm described in arXiv:1711.10409 that were needed to do the computation of the Tamagawa numbers and the real period in reasonable time. We also explain some of the more technical details of the algorithm, and give a brief overview of the methods used to compute the special value of the $L$-function and the regulator.Comment: Source code included as ancillary files. Comments always welcome

Explicit arithmetic intersection theory and computation of Néron-Tate heights

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over Q, and we show how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares

Explicit arithmetic intersection theory and computation of Néron-Tate heights

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over Q, and we show how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares