Computing Néron-Tate heights of points on hyperelliptic Jacobians

Abstract

It was shown by Faltings and Hriljac 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 hyperelliptic curves. To demonstrate the practicality of our algorithm, we illustrate it by computing Néron-Tate heights on Jacobians of hyperelliptic curves of genus from 1 to 9.

13 pages. v5: As kindly pointed out by Raymond van Bommel, the height is computed in this paper with respect to twice the theta divisor, not the theta divisor itself (as written in v4)

References in corpus (1)

• Computing canonical heights using arithmetic intersection theory

Cited by in corpus (8)

• An explicit theory of heights for hyperelliptic Jacobians of genus three
• Canonical Heights on Genus Two Jacobians
• An Arakelov-Theoretic Approach to Naïve Heights on Hyperelliptic Jacobians
• Explicit Kummer varieties of hyperelliptic Jacobian threefolds
• Numerical verification of the Birch and Swinnerton-Dyer conjecture for hyperelliptic curves of higher genus over $\mathbb{Q}$ up to squares
• Canonical heights and division polynomials
• Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction
• Explicit arithmetic intersection theory and computation of Néron-Tate heights