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Computing the canonical height on K3 surfaces

Authors:	Gregory S Call Joseph H Silverman

Institution:	address Department of Mathematics and Computer Science, Amherst College, Amherst, Massachusetts 01002 ; address Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912

Abstract:	Let 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 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 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 $\hat{h} ^+$ , $\hat{h} ^-$ .

Keywords:	K3 surface canonical height

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