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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
Growth of balls of holomorphic sections and energy at equilibrium
, 2008
"... Let L be a big line bundle on a compact complex manifold X. Given a nonpluripolar compact subset K of X and the weight φ of a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K, φ) as the AubinMabuchi energy of the extremal psh weight associated to (K, φ). We prove t ..."
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Cited by 42 (10 self)
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Let L be a big line bundle on a compact complex manifold X. Given a nonpluripolar compact subset K of X and the weight φ of a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K, φ) as the AubinMabuchi energy of the extremal psh weight associated to (K, φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k → ∞ of the volume of the supnorm unit ball induced by (K, kφ) on the space of global holomorphic sections H 0 (X, kL). As a consequence of these results, we recover and extend Rumely’s Robintype formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.
Arithmetic height functions over finitely generated fields
 Inventiones Mathematicae 140
, 2000
"... ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS ..."
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Cited by 37 (10 self)
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ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS
Topological Tits alternative
 the Annals of Math
, 2004
"... Abstract. Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that either Γ contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups. 1. ..."
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Cited by 32 (12 self)
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Abstract. Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that either Γ contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups. 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
The Bogomolov conjecture for totally degenerate abelian varietieties
"... Let K = k(B) be a function field of an integral projective variety B over the algebraically closed field k such that B is regular in codimension 1. The set of ..."
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Cited by 19 (4 self)
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Let K = k(B) be a function field of an integral projective variety B over the algebraically closed field k such that B is regular in codimension 1. The set of
Equidistribution of CMpoints on quaternion Shimura varieties
"... The aim of this paper is to show some equidistribution statements of Galois orbits of CMpoints for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CMpoints as predicted by AndréOort conjecture (see Section 2). Our main result (Corollary 3.7) say ..."
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Cited by 16 (1 self)
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The aim of this paper is to show some equidistribution statements of Galois orbits of CMpoints for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CMpoints as predicted by AndréOort conjecture (see Section 2). Our main result (Corollary 3.7) says that the Galois orbits of CMpoints with the maximal
The MordellLang Theorem for Drinfeld modules
 Internat. Math. Res. Notices
"... Faltings proved the MordellLang conjecture in the following form (see [6]). ..."
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Cited by 13 (9 self)
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Faltings proved the MordellLang conjecture in the following form (see [6]).
EQUIDISTRIBUTION AND GENERALIZED MAHLER MEASURES
"... Abstract. If K is a number field and ϕ: P 1 K − → P 1 K is a rational map of degree d> 1, then at each place v of K, one can associate to ϕ a generalized Mahler measure for polynomials F ∈ K[t]. These Mahler measures give rise to a formula for the canonical height hϕ(β) of an element β ∈ K; this ..."
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Abstract. If K is a number field and ϕ: P 1 K − → P 1 K is a rational map of degree d> 1, then at each place v of K, one can associate to ϕ a generalized Mahler measure for polynomials F ∈ K[t]. These Mahler measures give rise to a formula for the canonical height hϕ(β) of an element β ∈ K; this formula generalizes Mahler’s formula for the usual Weil height h(β). In this paper, we use diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging log F v over the periodic points of ϕ. This paper is dedicated to the memory of Serge Lang, who taught the world number theory for more than fifty years, through his research, lectures, and books. The usual Weil height of a rational number x/y, where x and y are integers without a common prime factor, is defined as h(x/y) = log max(x, y). More generally, one can define the usual Weil height h(β) of an algebraic number β in a number field K by summing log max(βv, 1) over all of the absolute values v of K. Mahler ([Mah60]) has proven that if F is a nonzero irreducible polynomial in Z[t] with coprime coefficients such that F (β) = 0, then ∫ 1 (0.0.1) deg(F)h(β) = log F (e 2πiθ)dθ.