arXiv:1207.5948 An Arakelov-Theoretic Approach to Naïve Heights on Hyperelliptic Jacobians fra arXiv Front: math.NT av David Holmes We give an Arakelov-theoretic definition of a naïve height on divisors of degree zero on a hyperelliptic curve over a number field, and show that this naïve height has computably bounded difference from the Néron-Tate height of the corresponding point on the Jacobian, a key ingredient being a theorem of Faltings and Hriljac comparing the Néron-Tate height on the Jacobian to the arithmetic intersection pairing on the curve. We then use this result to give a new algorithm to compute the finite set of points on a hyperelliptic Jacobian of Néron-Tate height less than a given bound. This has applications to the problem of saturation, to the computation of integral points on hyperelliptic curves, to the use of Manin’s algorithm, and for numerically testing cases of the Conjecture of Birch and Swinnerton-Dyer.
In Soule http://www.ams.org/mathscinet-getitem?mr=1144338 there is an overview of results from Arakelov theory and a comparison of the Faltings and Philippon height, the latter based on the Chow form. Late 80s.
arXiv:0907.1458 Theta height and Faltings height from arXiv Front: math.NT by F. Pazuki Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized abelian variety. We also give as an application an explicit upper bound on the number of K-rational points of a curve of genus g>1 over a number filed K under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.
arXiv:1001.2517 Heights and measures on analytic spaces. A survey of recent results, and some remarks from arXiv Front: math.NT by Antoine Chambert-Loir This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.
arXiv:1004.4503 Computing Néron-Tate heights of points on hyperelliptic Jacobians from arXiv Front: math.NT by David Holmes 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.
nLab page on Heights