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91
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
Harmonic analysis on metrized graphs
 CANAD. J. MATH
"... This paper studies the Laplacian operator on a metrized graph, and its spectral theory. ..."
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Cited by 38 (6 self)
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This paper studies the Laplacian operator on a metrized graph, and its spectral theory.
Effectivity of Arakelov Divisors and the Theta Divisor of a Number
"... In the well known analogy between the theory of function fields of curves over finite fields and the arithmetic of algebraic number fields, the number theoretical analogue of a divisor on a curve is an Arakelov divisor. In this paper we introduce the notion of an effective ..."
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Cited by 21 (1 self)
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In the well known analogy between the theory of function fields of curves over finite fields and the arithmetic of algebraic number fields, the number theoretical analogue of a divisor on a curve is an Arakelov divisor. In this paper we introduce the notion of an effective
ON THE ARAKELOV GEOMETRY OF MODULI SPACES OF CURVES
, 2002
"... In this paper we consider some problems in the Arakelov geometry of Mg, the moduli space of smooth projective curves of genus g over C. Specifically, we are interested in naturally metrized line bundles over Mg and their extensions to Mg, the DeligneMumford compactification of Mg. These line bundle ..."
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Cited by 17 (3 self)
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In this paper we consider some problems in the Arakelov geometry of Mg, the moduli space of smooth projective curves of genus g over C. Specifically, we are interested in naturally metrized line bundles over Mg and their extensions to Mg, the DeligneMumford compactification of Mg. These line bundles typically occur
CONTINUITY OF VOLUMES ON ARITHMETIC VARIETIES
, 2006
"... ABSTRACT. We introduce the volume function for C ∞hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic HilbertSamuel formula ..."
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Cited by 15 (9 self)
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ABSTRACT. We introduce the volume function for C ∞hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic HilbertSamuel formula for small sections of higher multiples of a nef C ∞hermitian invertible sheaf.