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68
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
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.
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.
Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves
 J. Amer. Math. Soc
, 1998
"... Abstract. Let f: X → Y be a surjective and projective morphism of smooth quasiprojective varieties over an algebraically closed field of characteristic zero with dimf = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if Xy is smooth and Ey is semistable for some ..."
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Cited by 26 (3 self)
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Abstract. Let f: X → Y be a surjective and projective morphism of smooth quasiprojective varieties over an algebraically closed field of characteristic zero with dimf = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if Xy is smooth and Ey is semistable for some y ∈ Y, then f ∗ 2rc2(E) − (r − 1)c1(E) 2) is weakly positive at y. We apply this result to obtain the following description of the cone of weakly positive QCartier divisors on the moduli space of stable curves. Let Mg (resp. Mg) be the moduli space of stable (resp. smooth) curves of genus g ≥ 2. Let λ be the Hodge class and δi’s (i = 0,..., [g/2]) the boundary classes. Then, a QCartier divisor xλ+ ∑ [g/2] i=0 yiδi on Mg is weakly positive over Mg if and only if x ≥ 0, gx+(8g +4)y0 ≥ 0, and i(g − i)x + (2g + 1)yi ≥ 0 for all 1 ≤ i ≤ [g/2].
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.
A Combination of the Conjectures by MordellLang and AndréOort
 In: Geometric Methods in Algebra and Number Theory (Bogomolov, F., Tschinkel, Y., Eds.) Progress in Math. 235
, 2005
"... Summary. We propose a conjecture combining the Mordell–Lang conjecture with an important special case of the André–Oort conjecture, and explain how existing results imply evidence for it. 1 ..."
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Cited by 24 (0 self)
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Summary. We propose a conjecture combining the Mordell–Lang conjecture with an important special case of the André–Oort conjecture, and explain how existing results imply evidence for it. 1
Ominimality and the André–Oort conjecture for Cn
 Annals of Mathematics. Second Series
, 2011
"... We give an unconditional proof of the AndréOort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the ManinMumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in power ..."
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Cited by 21 (2 self)
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We give an unconditional proof of the AndréOort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the ManinMumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in powers of the multiplicative group. The second includes the ManinMumford conjecture for abelian varieties defined over Q. Our approach uses the theory of ominimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the ManinMumford conjecture by PilaZannier; a proof of a special (but new) case of Pink’s relative ManinMumford conjecture by MasserZannier; and new proofs of certain known results of AndréOortManinMumford type by Pila. 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
Towards a dynamical ManinMumford conjecture
 Int. Math. Res. Not
"... Abstract. We provide a family of counterexamples to a first formulation of the dynamical ManinMumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of abelian varieties under the action of endomorphisms of abelian varieties and for lines under the act ..."
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Cited by 17 (8 self)
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Abstract. We provide a family of counterexamples to a first formulation of the dynamical ManinMumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of abelian varieties under the action of endomorphisms of abelian varieties and for lines under the action of diagonal endomorphisms of P 1 × P 1 .