Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing the canonical height on K3 surfaces

Authors: Gregory S. Call and Joseph H. Silverman
Journal: Math. Comp. 65 (1996), 259-290
MSC (1991): Primary 11G35, 11Y50, 14G25, 14J20, 14J28
MathSciNet review: 1322885
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Abstract: Let $S$ be a surface in $\mathbb P ^2\times \mathbb P ^2$ given by the intersection of a (1,1)-form and a (2,2)-form. Then $S$ is a K3 surface with two noncommuting involutions $\sigma ^x$ and $\sigma ^y$. In 1991 the second author constructed two height functions $\hat{h} ^+$ and $\hat{h} ^-$ which behave canonically with respect to $\sigma ^x$ and $\sigma ^y$, and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights $\sum _v\hat{\lambda} ^\pm (\,\cdot \,,v)$. We discuss how the geometry of the surface $S$ is related to formulas for the local heights, and we give practical algorithms for computing the involutions $\sigma ^x$, $\sigma ^y$, the local heights $\hat{\lambda} ^+(\,\cdot \,,v)$, $\hat{\lambda} ^-(\,\cdot \,,v)$, and the canonical heights $\hat{h} ^+$, $\hat{h} ^-$.

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Additional Information

Gregory S. Call
Affiliation: address Department of Mathematics and Computer Science, Amherst College, Amherst, Massachusetts 01002

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

Keywords: K3 surface, canonical height
Received by editor(s): August 2, 1994
Additional Notes: Research of the first author was partially supported by NSF ROA-DMS-8913113, NSA MDA 904-93-H-3022, and an Amherst Trustee Faculty Fellowship.
Research of the second author was partially supported by NSF DMS-9121727.
Article copyright: © Copyright 1996 American Mathematical Society