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46
Questions on self maps of algebraic varieties
 J. Ramanujan Math. Soc
"... In this note we shall consider some arithmetic and geometric questions concerning projective varieties X with a selfmap φ and an ample line bundle L such that φ ∗ (L) ∼ = L ⊗d for some d> 1, all defined over some field k. Such a situation is interesting arithmetically, if k is a number field, ..."
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Cited by 50 (1 self)
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In this note we shall consider some arithmetic and geometric questions concerning projective varieties X with a selfmap φ and an ample line bundle L such that φ ∗ (L) ∼ = L ⊗d for some d> 1, all defined over some field k. Such a situation is interesting arithmetically, if k is a number field, because using
Equirépartition des petits points
, 1997
"... Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont mêm ..."
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Cited by 42 (4 self)
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Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont même équidistribués
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
TOWARDS A DYNAMICAL MANINMUMFORD CONJECTURE
"... 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 endomorphisms of abelian varieties and for endomorphisms of P 1 × P 1. ..."
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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 endomorphisms of abelian varieties and for endomorphisms of P 1 × P 1.
Cohomological arithmetic Chow rings
, 2005
"... We develop a theory of abstract arithmetic Chow rings, where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of H. Gillet and C. Soulé ..."
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Cited by 14 (4 self)
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We develop a theory of abstract arithmetic Chow rings, where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of H. Gillet and C. Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with loglog singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.
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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Cited by 12 (2 self)
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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
DYNAMICS OF POLYNOMIAL MAPPINGS OF C2
"... Abstract. We study the dynamics of polynomial self mappings f of C2. We construct, for a large class of mappings, an invariant measure µ which is mixing and of maximal entropy hµ ( f) = ..."
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Cited by 9 (0 self)
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Abstract. We study the dynamics of polynomial self mappings f of C2. We construct, for a large class of mappings, an invariant measure µ which is mixing and of maximal entropy hµ ( f) =
The Canonical Arithmetic Height Of Subvarieties Of An Abelian Variety Over A Finitely Generated Field
, 1999
"... This paper is the sequel of [2]. In [4], S. Zhang defined the canonical height of subvarieties of an abelian variety over a number field in terms of adelic metrics. In this paper, we generalize it to an abelian variety defined over a finitely generated field over Q.Ourwayis ..."
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Cited by 8 (3 self)
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This paper is the sequel of [2]. In [4], S. Zhang defined the canonical height of subvarieties of an abelian variety over a number field in terms of adelic metrics. In this paper, we generalize it to an abelian variety defined over a finitely generated field over Q.Ourwayis
Equidistribution of dynamically small subvarieties over the function field of a curve
 Acta Arith
"... Abstract. A morphism ϕ: X → X of a projective variety defined over a field K is called an algebraic dynamical system if it has a special kind of polarization. In this paper we take K to be the function field of a smooth curve and prove that at each place v of K, subvarieties of dynamically small hei ..."
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Cited by 6 (0 self)
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Abstract. A morphism ϕ: X → X of a projective variety defined over a field K is called an algebraic dynamical system if it has a special kind of polarization. In this paper we take K to be the function field of a smooth curve and prove that at each place v of K, subvarieties of dynamically small height are equidistributed on the associated Berkovich analytic space. We carefully develop all of the arithmetic intersection theory needed to state and prove this theorem, and we present several applications on the nonZariski density of preperiodic points and of points of small height in field extensions of bounded degree. X an v Contents