Computing N\'eron-Tate heights of points on hyperelliptic Jacobians

TL;DR

This paper presents an explicit algorithm for computing Néron-Tate heights of points on hyperelliptic Jacobians, enabling practical calculations for curves of genus 1 to 9.

Contribution

It makes the theoretical height formula explicit and develops a practical algorithm for hyperelliptic Jacobians, demonstrated on curves of various genera.

Findings

Successfully computed Néron-Tate heights for hyperelliptic Jacobians of genus 1 to 9

Algorithm is practical and effective for a range of hyperelliptic curves

Provides a computational tool bridging theory and explicit height calculations

Abstract

It was shown by Faltings and Hriljac that the N\'eron-Tate height of a point on the Jacobian of a curve can be expressed as the self-intersection of a corresponding divisor on a regular model of the curve. We make this explicit and use it to give an algorithm for computing N\'eron-Tate heights on Jacobians of hyperelliptic curves. To demonstrate the practicality of our algorithm, we illustrate it by computing N\'eron-Tate heights on Jacobians of hyperelliptic curves of genus from 1 to 9.