Dynamics of Projective Morphisms Having Identical Canonical Heights

Let φ,ψ : PN → PN be morphisms of degree at least 2 whose canonical heights ĥφ and ĥψ are identical. We draw various conclusions about the Green functions, Julia sets, and canonical local heights of φ and ψ. We use this information to completely characterize φ and ψ in the following cases: (i) φ and ψ are polynomial maps in one variable; (ii) φ is the dth-power map; (iii) φ is a Lattès map. Introduction Let φ : P → P be a morphism of degree d 2 defined over Q̄ and let h : P (Q̄) → R be the usual Weil height on P . The (global) canonical height associated to φ is the function ĥφ : P (Q̄) −→ R, ĥφ(P ) = lim n→∞ 1 dn h ( φ(P ) ) . It has many interesting dynamical and arithmetic properties, including ĥφ(P ) = 0 ⇐⇒ P is preperiodic for φ. An immediate consequence is Northcott’s theorem [21] that φ has only finitely many preperiodic points defined over all number fields of bounded degree. In this paper we investigate what can be deduced about two morphisms φ and ψ whose canonical height functions ĥφ and ĥψ coincide. Aside from its intrinsic interest, this paper represents the initial part of a project to use the sup norm on the space of canonical heights as a means of measuring the arithmetic distance between morphisms. Thus if we define ArithDist(φ,ψ) = sup P∈PN (Q̄) ∣∣ĥφ(P )− ĥψ(P )∣∣, then our results characterize morphisms whose arithmetic distance from one another is zero. Further properties of the arithmetic distance function are developed in [14]. We collect our main results into two theorems. Theorem 1. Let φ and ψ be morphisms P → P of degree at least 2. The condition ĥφ = ĥψ has the following consequences. (a) The associated Green functions Gφ( · , v) = Gψ( · , v) are equal (after appropriate normalization). (b) Fix an embedding Q̄ ⊂ C. Then the Julia sets J (φ) = J (ψ) in P (C) coincide. (c) Let φ and ψ be polynomial maps, that is, maps that induce morphisms A → A , where A = P \ {x0 = 0}. Then the normalized local heights λ̂φ = λ̂ψ coincide. Received 6 September 2006; revised 7 February 2007; published online 26 June 2007. 2000 Mathematics Subject Classification 11G50 (primary), 14G40, 58F23 (secondary). The first author’s research was supported by MEXT grant-in-aid for young scientists (B) 18740008. The second author’s research was supported by NSA grant H98230-04-1-0064. 520 SHU KAWAGUCHI AND JOSEPH H. SILVERMAN (d) Let φ,ψ ∈ Q̄[x] be polynomial maps in one variable. Then after a simultaneous linear conjugation and up to multiplication by appropriate roots of unity, one of the following is true: (i) φ and ψ are both powers of x; (ii) φ and ψ are both Tchebycheff polynomials; (iii) φ and ψ are both iterates of a common polynomial f . Our other main result deals with the situation in which one of the morphisms is associated to an endomorphism of an algebraic group. Theorem 2. (a) Let φ : P1 → P1 be a Lattès map on P1, that is, φ is associated to an endomorphism of an elliptic curve E (see [20]). Suppose that ψ : P1 → P1 satisfies ĥψ = ĥφ. Then ψ is also a Lattès map attached to E. (b) Let φ : P → P be a morphism of degree at least 2 whose canonical height ĥφ is equal to the classical Weil height h. Then φ has the form (ξ0X i0 : ξ1X d i1 : . . . : ξNX d iN ) for a permutation (i0, . . . , iN ) of {0, 1, . . . , N} and roots of unity ξ0, . . . , ξN . Equivalently, φ is a composition of an endomorphism of Gm with translation by a torsion point. We briefly describe the three main sections of the paper. Section 1. Green functions and height functions. In the first section we develop a theory of Green functions attached to morphisms φ : P → P and explain how these Green functions are related to both local and global canonical height functions. We expect that this machinery will find many uses in the study of arithmetic dynamics, as for example is shown in the onevariable case by Baker and Rumeley [3]. Indeed, much of the material in the first part of this paper may be viewed as generalizing the more elementary parts of [3] to higher dimensions. See Section 1.6 for further details. Section 2. Morphisms with identical canonical heights. In this section we use the Green function/canonical height machinery developed in Section 1 to analyze morphisms whose canonical heights are the same. A key result is Theorem 20, which says that if the global canonical heights ĥφ and ĥψ coincide, then the associated Green functions GΦ( · ; v) and GΨ( · ; v) are essentially identical. The proof uses properties of Green functions proven in Section 1 and the weak approximation theorem in order to break out individual terms from the sum ∑ v GΦ(x; v). We note that the section starts quite generally as in Theorem 1(a)(b). Next we specialize to polynomial maps as in Theorem 1(c), which lets us relate Green functions to canonical local heights. Finally, we end the section by restricting to polynomial maps in one variable as in Theorem 1(d), which allows us to apply classical results from dynamics over C describing maps whose Julia sets coincide. Section 3. Canonical heights and algebraic groups. The final part deals with morphisms associated to endomorphisms of algebraic groups G, more specifically endomorphisms of tori Gm ⊂ P and of elliptic curves E. The group structure on G provides a strong tool for analyzing dynamics, and we prove something considerably stronger than the results stated in Theorem 2. If two morphisms φ and ψ have identical canonical heights, then their sets of preperiodic points coincide, so it is natural to ask the following purely geometric question: if PrePer(φ) = PrePer(ψ), then what can be said about φ and ψ? We consider the following question with an even weaker hypothesis. DYNAMICS AND IDENTICAL CANONICAL HEIGHTS 521 Question 3. Let φ,ψ : P → P be morphisms of degree at least 2. Suppose that there is a Zariski dense set of points Z ⊂ P satisfying Z ⊂ PrePer(φ) and ψ(Z) ⊂ PrePer(φ). (1) What can be said about the relationship between φ and ψ? Our main results in Section 3 (Theorems 27 and 34) say that if φ is associated to an endomorphism of certain algebraic groups G and if (1) is true, then ψ is also associated to an endomorphism of G composed with translation by an element of Gtors. (More precisely, we prove this if G is an elliptic curve E, that is, φ is a Lattès map, or if G is the torus Gm.) The proofs of Theorems 27 and 34 rely on deep arithmetic theorems of Raynaud and Laurent, respectively, describing which subvarieties of algebraic groups contain large numbers of torsion points. Remark 4. Dvornicich and Zannier [8] have recently studied the question of which polynomial maps φ have the property that PrePer(φ) contains infinitely many, or a Zariski dense set of, points whose coordinates are roots of unity. Their results are thus related to our Theorem 2(a) (and Theorem 34), and they, too, use Laurent’s theorem [18] as a key tool in some of their proofs. Remark 5. Our results assume that ĥφ = ĥψ for all points in P (Q̄). It is interesting to ask whether the results remain true if ĥφ and ĥψ are only required to agree on a smaller set of points. For example, suppose that φ and ψ are defined over Q and that ĥφ(P ) = ĥψ(P ) for all Q-rational points P ∈ P (Q). What can be said about φ and ψ? We do not know the answer even in the case that ĥψ is the usual Weil height function h. Acknowledgements. The authors would like to thank Rob Benedetto for his helpful suggestions and Umberto Zannier for sending us a preprint of his paper [8]. The first author would also like to thank the Institut de Mathématiques de Jussieu and Vincent Maillot for their warm hospitality. 1. Green functions and height functions We begin by setting some notation that will remain in effect throughout this paper. If K is any field and | · | an absolute value on K, we extend | · | to vectors using the sup norm, ∥∥(x0, . . . , xN )∥∥ = max{|x0|, . . . , |xN |}. Let φ : P P be a rational map of degree d 2 defined over K. Then φ can be written as φ = (Φ0 : . . . : ΦN ) with homogeneous polynomials Φ0, . . . ,ΦN ∈ K[x0, . . . , xN ] having no common factors. We call the induced map Φ = (Φ0, . . . ,ΦN ) : KN+1 −→ KN+1 a lift of φ. The lift is unique up to multiplication by a constant in K×. The rational map φ : P → P is a morphism if and only if the homogeneous polynomials Φ0, . . . ,ΦN have no non-trivial common zeros in K̄N+1. For notational convenience we write (KN+1)∗ = KN+1 \ {0}, and then the lift of a morphism is a map Φ : (KN+1)∗ −→ (KN+1)∗. 522 SHU KAWAGUCHI AND JOSEPH H. SILVERMAN We also let π : (KN+1)∗ −→ P (K) be the natural projection, so the lifts Φ of φ are characterized by the condition π ◦ Φ = φ ◦ π. 1.1. Elimination theory and the Macaulay resultant We briefly review some facts from elimination theory. We state only what we need and refer the reader to [5, 10, 11, 13, 26] for general results and further properties of the resultant. We set the following notation: A a ring (commutative with 1); m the ideal (x0, . . . , xN ) in A[x0, . . . , xN ]; f0, . . . , fr homogeneous polynomials in m; I(f0, . . . , fr) the ideal of A[x0, . . . , xN ] generated by (f0, . . . , fr). Associated to the coefficient ring A and the polynomials f0, . . . , fr is an ideal RA(f0, . . . , fr) of A that may be defined in a number of equivalent ways, including the following: RA(f0, . . . , fr) = { t ∈ A : there is a k 1 such that mt ⊂ I(f0, . . . , fr) } . The fundamental theorem of elimination theory says that specializations of RA(f0, . . . , fr) determine whether f0, . . . , fr have a non-trivial common zero. Theorem 6 (Elimination Theory). Let K be a field, let K̄ be an algebraic closure of K, and let ρ : A → K be a homomorphism. We extend ρ to A[x0, . . . , xN ] by applying ρ to the coefficients of a polynomial. Then the following are equivalent: (a) ρ ( RA(f0, . . . , fr) ) = (0); (b) there is a point P ∈ P (K̄) such that ρ(f0)(P ) = ρ(f1)(P ) = . . . = ρ(fr)(P ) = 0. Proof. See [10, II.4.9], [11, I.5.7A] or [26, II, § 80]. Now let A be the universal coefficient ring for homogeneous polynomials of given degrees d0, . . . , dr 0. Thus for each 0 i r we write fi(x0, . . . , xN ) = ∑ α0+...+αN=di Ti,αx α0 0 . . . x αN N using algebraically independent coefficients Ti,α, and we let A = Z[Ti,α] be the Z-algebra generated by the coefficients of all of the fi. Theorem 7 (Macaulay Resultant). Let A be the universal coefficient ring described above. Assume that r = N . Then RA(f0, . . . , fN ) is a principal and prime ideal of A. It has a unique generator, which we denote Res(f0, . . . , fN ), characterized by the property Res(x0 0 , . . . , x dN N ) = 1 ∈ Z. We call Res(f0, . . . , fN ) ∈ A the Macaulay resultant of f0, . . . , fN . Proof. This is a classical result. See [5, Theorem 3.8], [7, Chapter 3] or [13] for a proof. We use the Macaulay resultant to relate the size of x to the size of Φ(x). This generalizes [3, Lemma 3.1 and Remark] from P1 to P . The proof of part (a) actually only requires the Nullstellensatz, but (b) gives more precise information via the Macaulay resultant. DYNAMICS AND IDENTICAL CANONICAL HEIGHTS 523 Proposition 8. Let (K, v) be a valued field, let φ : P → P be a morphism of degree d 2, and let Φ = (Φ0, . . . ,ΦN ) : (KN+1)∗ −→ (KN+1)∗ be a lift of φ. (a) There are constants c1, c2 > 0, depending on Φ, K, and v, so that c1‖x‖v ‖Φ(x)‖v c2‖x‖v for all x ∈ (KN+1)∗. (b) Suppose v is non-archimedean and that the coefficients of the polynomials Φ0, . . . ,ΦN are v-integral. Let Res(Φ) denote the Macaulay resultant of Φ0, . . . ,ΦN . Then |Res(Φ)|v · ‖x‖v ‖Φ(x)‖v ‖x‖v for all x ∈ (KN+1)∗. Proof. The upper bound is an elementary consequence of the triangle inequality. Thus if we write Φi(x0, . . . , xN ) = ∑ α0+...+αN=d ai,αx α0 0 . . . x αn n , then for any point x ∈ (KN+1)∗ we have ∥∥Φ(x0, . . . , xN )∥∥v = max 0 i N∣∣Φi(x0, . . . , xN )∣∣v c3(N, d)max i,α |ai,α|v · ‖x‖v. This gives the upper bound in (a). Further, if v is non-archimedean, then we can take c3(N, d) = 1; and if further the coefficients of the Φi are v-integral, then each |ai,α|v 1, which gives the upper bound in (b). To prove the lower bound, we use elimination theory. Let A be the ring generated over Z by the coefficients of Φ0, . . . ,ΦN . Theorem 7 says that the ideal R(Φ0, . . . ,ΦN ) ⊂ A is generated by Res(Φ), so the definition of R(Φ0, . . . ,ΦN ) gives us an integer k 1 such that (x0, . . . , xN ) · Res(Φ) ⊂ I(Φ0, . . . ,ΦN ). In particular, for each 0 i N we can find homogeneous polynomials Ψij ∈ A[x0, . . . , xN ] satisfying