Canonical Height Functions Defined on the Affine Plane Associated with Regular Polynomial Automorphisms

Let f : A → A be a regular polynomial automorphism (e.g., a Hénon map) defined over a number field K. We construct canonical height functions defined on A(K) associated with f . These functions satisfy the Northcott finiteness property, and an Kvalued point on A(K) is f -periodic if and only if its height is zero. As an application of canonical height functions, we give a refined estimate on the number of points with bounded height in an infinite f -orbit. Introduction and the statement of the main results One of the basic tools in Diophantine geometry is the theory of height functions. On Abelian varieties defined over a number field, Néron and Tate developed the theory of canonical height functions that behave well relative to the [n]-th map (cf. [8, Chap. 5]). On certain K3 surfaces with two involutions, Silverman [12] developed the theory of canonical height functions that behave well relative to the two involutions. For the theory of canonical height functions on some other projective varieties, see for example [1], [14], [6]. In this paper, we construct canonical height functions defined on the affine plane, which behave well relative to regular polynomial automorphisms, and in particular Hénon maps. A Hénon map (also called a generalized Hénon map) is a polynomial automorphism f : A → A of the form (0.1) f ( x y ) = ( p(x)− ay x ) , where a 6= 0 and p is a polynomial of degree d ≥ 2. Hénon maps are basic objects in polynomial automorphisms of A in the sense that every polynomial automorphism of A of degree d ≥ 2 over C is conjugate to either an elementary map, or a composite of Hénon maps (Friedland–Milnor [3]). A regular polynomial automorphism f : A → A is by definition a polynomial automorphism of A of degree greater than or equal to 2 such that the unique point of indeterminacy of f is different from the the unique point of indeterminacy of f−1, where the birational map f : P · · · → P (resp. f−1 : P · · · → P) is the extension of f (resp. f). Hénon maps are examples of regular polynomial automorphisms. For more details, see the survey of Sibony [10] and the references therein. Over a number field, Silverman [13] studied arithmetic properties of quadratic Hénon maps, and then Denis [2] studied arithmetic properties of Hénon maps and some classes of polynomial automorphisms. 1991 Mathematics Subject Classification. 11G50.