Computing Néron–Tate heights of points on hyperelliptic Jacobians

It was shown by Faltings (1984) [Fal84], and Hriljac (1985) [Hri85] that the Néron–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 Néron–Tate heights on Jacobians of (hyper)elliptic curves. To demonstrate the practicality of our algorithm, we illustrate it by computing Néron–Tate heights on Jacobians of (hyper)elliptic curves of genus 1⩽g⩽9.