Canonical heights and division polynomials

TL;DR

This paper introduces a novel computational method for canonical heights on hyperelliptic Jacobians over number fields, avoiding complex models and factorization, especially effective in genus 2 cases.

Contribution

It presents a new approach based on division polynomials and Diophantine approximation, eliminating the need for geometrical models or factorization in certain cases.

Findings

Method efficiently computes canonical heights without geometric models.

Genus 2 case requires no factorization.

Applicable over number fields with no complex analytic tools.

Abstract

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither $p$-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the `division polynomials' associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.