Computing canonical heights using arithmetic intersection theory

TL;DR

This paper presents a practical method to compute canonical heights on Jacobians of curves using arithmetic intersection theory, including a complete algorithm for hyperelliptic curves implemented in Magma.

Contribution

It introduces a new computational approach for canonical heights on Jacobians, with a complete algorithm for hyperelliptic curves and implementation details.

Findings

Algorithm successfully computes canonical heights in examples.

Implementation in Magma demonstrates practical feasibility.

Running time behavior is analyzed.

Abstract

For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.