Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two

Abstract

AbstractLet V be an algebraic K3 surface defined over a number field K. Suppose V has Picard number two and an infinite group of automorphisms A = Aut(V/K). In this paper, we introduce the notion of a vector height h: V → Pic(V) ⊗ and show the existence of a canonical vector height with the following properties:where σ ∈ A, σ* is the pushforward of σ (the pullback of σ−1), and hD is a Weil height associated to the divisor D. The bounded function implied by the O(1) does not depend on P. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an A-orbit satisfiesHere, μ(P) is a nonnegative integer, s is a positive integer, and ω is a real quadratic fundamental unit.