Explicit arithmetic intersection theory and computation of N\'eron-Tate heights

TL;DR

This paper introduces a general algorithm for computing intersection pairings on arithmetic surfaces, enabling the calculation of regulators and verification of the Birch and Swinnerton-Dyer conjecture for specific Jacobians.

Contribution

It presents a novel algorithm for intersection theory on arithmetic surfaces and demonstrates its application to compute regulators and verify BSD conjecture numerically.

Findings

Successfully computed regulators for Jacobians of smooth plane quartics.

Numerically verified the BSD conjecture for the Jacobian of the split Cartan curve of level 13.

Implemented the algorithm for curves over 5, showing practical effectiveness.

Abstract

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb Q$, 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.