Computing Néron-Tate heights of points on hyperelliptic Jacobians. (English) Zbl 1239.14019
Let \(C\) be an odd-degree hyperelliptic curve over a number field. The aim of this paper is the practical computation of the Néron-Tate height of a point on the Jacobian of \(C\), by using Arakelov theory (more precisely, the formula of G. Faltings [Ann. Math. (2) 119, 387-424 (1984; Zbl 0559.14005)] and P. Hriljac [Am. J. Math. 107, 23–38 (1985; Zbl 0593.14004)], that expresses this height as the self-intersection of a certain divisor on a regular model of the curve).
Algorithms to compute the different ingredients of that formula are described, and some numerical tests for genus \(1\leq g\leq9\) are presented. For genus \(g>2\) this method seems much more efficient than previous attempts to compute Néron-Tate heights, based on explicit equations for projective embeddings of Jacobians.
Algorithms to compute the different ingredients of that formula are described, and some numerical tests for genus \(1\leq g\leq9\) are presented. For genus \(g>2\) this method seems much more efficient than previous attempts to compute Néron-Tate heights, based on explicit equations for projective embeddings of Jacobians.
Reviewer: Enric Nart Viñals (Barcelona)
MSC:
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11G50 | Heights |
| 37P30 | Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems |
Software:
MagmaReferences:
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