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93
Equidistribution of small points, rational dynamics, and potential theory
 Ann. Inst. Fourier (Grenoble
, 2006
"... Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points ..."
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Cited by 46 (7 self)
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Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕcanonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the wellknown canonical measure associated to ϕ. This theorem generalizes a result of BakerHsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable ArakelovGreen’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potentialtheoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the vadic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), who ..."
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Cited by 45 (14 self)
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It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6.
Intersections of polynomial orbits, and a dynamical MordellLang conjecture
 INVENT. MATH
, 2007
"... We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the MordellLang conjecture. ..."
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Cited by 29 (12 self)
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We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the MordellLang conjecture.
Canonical heights, transfinite diameters, and polynomial dynamics
 J. Reine Angew. Math
"... Abstract. Let φ(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating φ gives rise to a dynamical system and a corresponding canonical height function ˆ hφ, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of th ..."
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Cited by 26 (6 self)
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Abstract. Let φ(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating φ gives rise to a dynamical system and a corresponding canonical height function ˆ hφ, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of φ over various completions of K, and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero. 1.
MordellLang plus Bogomolov
 Invent. Math
, 1999
"... Let k be a number field. Let A be an almost split semiabelian variety over k; by this we mean that A is isogenous to the product of an abelian variety A0 and a torus T. We enlarge k if necessary to assume that T ∼ = Gn m. Let φ = (φ1, φ2) : A → A0 × Gn m be the isogeny. Let h1: A0(k) → R be a Néro ..."
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Cited by 19 (4 self)
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Let k be a number field. Let A be an almost split semiabelian variety over k; by this we mean that A is isogenous to the product of an abelian variety A0 and a torus T. We enlarge k if necessary to assume that T ∼ = Gn m. Let φ = (φ1, φ2) : A → A0 × Gn m be the isogeny. Let h1: A0(k) → R be a NéronTate canonical height associated to a symmetric ample line bundle on A0, and let h2: Gn m(k) → R be the sum of the naive heights of the coordinates. For x ∈ A(k), let h(x) = h1(φ1(x)) + h2(φ2(x)). For ǫ ≥ 0, let Bǫ = { z ∈ A(k)  h(z) ≤ ǫ}. Let Γ be a finitely generated subgroup of A(k), and define Γǫ: = Γ + Bǫ = { γ + z  γ ∈ Γ, h(z) ≤ ǫ}. Note that Γ0 = Γ + A(k)tors. Let X be a geometrically integral closed subvariety of A. Our main result is the existence of ǫ> 0 such that X(k) ∩ Γǫ is contained in a finite union ⋃ Zj where each Zj is a translate of a subsemiabelian variety of A k = A ⊗k k by a point in Γ0 and Zj ⊆ X
Heights and preperiodic points of polynomials over function fields, 2005. preprint. Available at arxiv:math.NT/0510444
"... Abstract. Let K be a function field in one variable over an arbitrary field F. Given a rational function φ ∈ K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of φ all have canonical height zero; conversely, ..."
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Abstract. Let K be a function field in one variable over an arbitrary field F. Given a rational function φ ∈ K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of φ all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be nonpreperiodic points of canonical height zero. In this paper, we show that for polynomial φ, such points exist only if φ is isotrivial. In fact, such Krational points exist only if φ is defined over the constant field of K after a Krational change of coordinates. Let K be a field with algebraic closure ˆ K, and let φ: P 1 ( ˆ K) → P 1 ( ˆ K) be a morphism defined over K. We may write φ as a rational function φ ∈ K(z). Denote the n th iterate of φ under composition by φ n. That is, φ 0 is the identity function, and for n ≥ 1, φ n = φ ◦φ n−1. A point x ∈ P 1 ( ˆ K) is said to be preperiodic under φ if there are integers n> m ≥ 0 such that φ m (x) = φ n (x). Note that x is preperiodic if and only if its forward orbit {φ n (x) : n ≥ 0} is finite. If K is a number field or a function field in one variable, and if deg φ ≥ 2, there is a
Primitive divisors in arithmetic dynamics
 Math. Proc. Camb. Phil. Soc
"... Abstract. Let ϕ(z) ∈ Q(z) be a rational function of degree at least 2 satisfying ϕ(0) = 0 and 0 ̸ = ϕ ′ (0) ∈ Z. Let α ∈ Q have infinite orbit under iteration of ϕ and write ϕ n (α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primi ..."
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Cited by 15 (3 self)
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Abstract. Let ϕ(z) ∈ Q(z) be a rational function of degree at least 2 satisfying ϕ(0) = 0 and 0 ̸ = ϕ ′ (0) ∈ Z. Let α ∈ Q have infinite orbit under iteration of ϕ and write ϕ n (α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p  An and p ∤ Ai for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a periodic point for ϕ.
Canonical height functions for affine plane automorphisms
, 2006
"... Abstract. Let f: A 2 → A 2 be a polynomial automorphism of dynamical degree δ ≥ 2 over a number field K. (This is equivalent to say that f is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on A 2 (K) associated with f. These functions sa ..."
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Abstract. Let f: A 2 → A 2 be a polynomial automorphism of dynamical degree δ ≥ 2 over a number field K. (This is equivalent to say that f is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on A 2 (K) associated with f. These functions satisfy the Northcott finiteness property, and an Kvalued point on A 2 (K) is fperiodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height in an infinite forbit. 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 power map (cf. [9, Chap. 5]). On certain K3 surfaces with two involutions, Silverman [14] developed the theory of canonical
LOCAL AND GLOBAL CANONICAL HEIGHT FUNCTIONS FOR AFFINE SPACE REGULAR AUTOMORPHISMS
, 909
"... ABSTRACT. Let f: A N → A N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the vadic Green functions Gf,v and G f −1,v (i.e., the vadic canonical height functions) for f and f −1. Next we introduce for f the notion of good reduction at v, and ..."
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Cited by 11 (1 self)
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ABSTRACT. Let f: A N → A N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the vadic Green functions Gf,v and G f −1,v (i.e., the vadic canonical height functions) for f and f −1. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of vadic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcotttype finiteness property. Using [7], we recover results on arithmetic properties of fperiodic points and non fperiodic points. We also obtain an estimate of growth of heights under f and f −1, which is independently obtained by Lee by a different method.