Local and global canonical height functions for affine space regular automorphisms

Let f : A → A be a regular polynomial automorphism defined over a number field K . For each place v of K , we construct the v-adic Green functions Gf,v and Gf−1,v (i.e., the v-adic canonical height functions) for f and f . Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using [7], we recover results on arithmetic properties of f -periodic points and non f -periodic points. We also obtain an estimate of growth of heights under f and f, which is independently obtained by Lee by a different method. INTRODUCTION Height functions are one of the basic tools in Diophantine geometry. On Abelian varieties defined over a number field, there exist Néron–Tate’s canonical height functions that behave well relative to the [n]-th power map. Tate’s elegant construction is via a global method using a relation of an ample divisor relative to the [n]-th power map. Néron’s construction is via a local method, and gives deeper properties of the canonical height functions. Both constructions are useful in studying arithmetic properties of Abelian varieties. In [7], we showed the existence of canonical height functions for affine plane polynomial automorphisms of dynamical degree ≥ 2. Our construction was via a global method using the effectiveness of a certain divisor on a certain rational surface. In this paper, we use a local method to construct a canonical height function for affine space regular automorphism f : A → A , which coincides with the one in [7] when N = 2. We note that arithmetic properties of polynomial automorphisms over number fields have been studied, for example, by Silverman [17], Denis [5], Marcello [12, 13] and the author [7]. We recall the definition of regular polynomial automorphisms. Let f : A → A be a polynomial automorphism of degree d ≥ 2 defined over a field, and f : P 99K P denote its birational extension to P . We write f for the inverse of f , d− for the degree of f, and f−1 for its birational extension to P . Then f is said to be regular if the intersection of the set of indeterminacy of f and that of f−1 is empty over an algebraic closure of the field (cf. Definition 2.1 and Remark 2.2). Over C, dynamical properties of affine space regular polynomial automorphisms f are deeply studied, in which the Green function for f plays a pivotal role (see [16, §2]). In §1 and §2, we construct a Green function (a local canonical height function) for f over an algebraically closed field Ω with non-trivial non-archimedean absolute value | · |. 2000 Mathematics Subject Classification: Primary: 11G50, Secondary: 32H50.