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Mathematics of Computation

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Computing Canonical Heights
With Little (or No) Factorization

Author: Joseph H. Silverman
Journal: Math. Comp. 66 (1997), 787-805
MSC (1991): Primary 11G05, 11Y50
MathSciNet review: 1388892
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Abstract: Let $E/ \mathbb {Q} $ be an elliptic curve with discriminant $\Delta $, and let $P\in E( \mathbb {Q})$. The standard method for computing the canonical height $ \hat h(P)$ is as a sum of local heights $\hat h (P)= \hat \lambda _{\infty }(P)+\sum _{p} \hat \lambda _{p}(P)$. There are well-known series for computing the archimedean height $\hat \lambda _{\infty }(P)$, and the non-archimedean heights $\hat \lambda _{p}(P)$ are easily computed as soon as all prime factors of $\Delta $ have been determined. However, for curves with large coefficients it may be difficult or impossible to factor $\Delta $. In this note we give a method for computing the non-archimedean contribution to $\hat h (P)$ which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.

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Additional Information

Joseph H. Silverman
Affiliation: Mathematics Department, Box 1917, Brown University, Providence, Rhode Island 02912

Keywords: Elliptic curve, canonical height
Received by editor(s): October 24, 1995
Additional Notes: Research partially supported by NSF DMS-9424642.
Article copyright: © Copyright 1997 American Mathematical Society