The Canonical Height on K 3 Surfaces

Let S be a surface in P2 × P2 given by the intersection of a (1,1)form and a (2,2)-form. Then S is a K3 surface with two noncommuting involutions σx and σy . In 1991 the second author constructed two height functions ĥ+ and ĥ− which behave canonically with respect to σx and σy , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights ∑ v λ̂ ±( · , 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 σx, σy , the local heights λ̂+( · , v), λ̂−( · , v), and the canonical heights ĥ+, ĥ−. Introduction Let S ⊂ P×P be a K3 surface defined by the vanishing of a (1,1)-form L(x,y) and a (2,2)-form Q(x,y). The two projections S → P are double covers, so they induce involutions σ, σ : S → S. The involutions σ and σ are rational maps, and they will be morphisms provided that the projections have no degenerate fibers, that is, no fibers of positive dimension. Suppose now that S is defined over a number field K and that σ, σ are morphisms. Then Silverman [6] has defined two height functions ĥ± : S(K̄) → [0,∞) which behave canonically relative to σ and σ. (See Theorem 3.1.) These heights have many interesting arithmetic properties, including the property that ĥ(P ) = 0⇐⇒ ĥ−(P ) = 0⇐⇒ P has finite orbit under σ and σ . Thus ĥ and ĥ− are analogous to the usual canonical heights on elliptic curves and abelian varieties. The construction of canonical heights can be extended to even more general settings whenever Tate’s telescoping sum construction applies, see [2, Theorem 1.1]. Néron and Tate have shown that the canonical height on an abelian variety can be decomposed into a sum of local height functions, one for each place of K, and this construction can also be generalized [2, Theorem 2.1]. The decomposition into local heights offers a more practical method for calculating the canonical height. For non-Archimedean v, one can show that if the variety Received by the editor August 2, 1994. 1991 Mathematics Subject Classification. Primary 11G35, 11Y50, 14G25, 14J20, 14J28.