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29
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
The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
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Cited by 39 (3 self)
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Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
REVIEW OF GEOMETRY AND ANALYSIS
, 2000
"... In this article, we shall discuss what the author considers to be important in geometry and related subjects. Since the time of the Greek mathematicians, geometry has always been in the center of science. Scientists cannot resist explaining natural phenomena in terms of the language of geometry. Ind ..."
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Cited by 19 (0 self)
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In this article, we shall discuss what the author considers to be important in geometry and related subjects. Since the time of the Greek mathematicians, geometry has always been in the center of science. Scientists cannot resist explaining natural phenomena in terms of the language of geometry. Indeed, it is reasonable to consider geometric objects as parts of nature. Practically all elegant theorems in geometry have found applications in classical or modern physics. In order to understand the future of geometry, it is perhaps useful to review what was known in the past. Clearly what I consider to be important may not be viewed to be so by others. Also, we should always keep in mind that what is fashionable now may not be so tomorrow. A theory can be judged to be successful only if its consequences help us understand the basic structure or the beauty of geometry. While we shall divide the subject into several categories, the division is artificial, as the development of each section depends on other sections heavily.
The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group
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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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Cited by 11 (2 self)
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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.
ARITHMETIC GEOMETRY OF TORIC VARIETIES. METRICS, MEASURES AND HEIGHTS
, 2011
"... We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider mo ..."
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Cited by 10 (3 self)
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We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real MongeAmpère measures, and LegendreFenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the FubiniStudy metric, and of some toric bundles.
MAHLER MEASURE FOR DYNAMICAL SYSTEMS ON P1 AND INTERSECTION THEORY ON A SINGULAR ARITHMETIC
"... Abstract. The Mahler measure formula expresses the height of an algebraic number as the integral of the log of the absolute value of its minimal polynomial on the unit circle. The height is in fact the canonical height associated to the monomial maps xn. We show in this work that for any rational ma ..."
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Cited by 5 (4 self)
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Abstract. The Mahler measure formula expresses the height of an algebraic number as the integral of the log of the absolute value of its minimal polynomial on the unit circle. The height is in fact the canonical height associated to the monomial maps xn. We show in this work that for any rational map ϕ(x) the canonical height of an algebraic number with respect to ϕ can be expressed as the integral of the log of its equation against the invariant BrolinLyubich measure associated to ϕ, with additional adelic terms at finite places of bad reduction. We give a complete proof of this theorem using integral models for each iterate of ϕ. In the last chapter on equidistribution and Julia, sets we give a survey of results obtained by P. Autissier, M. Baker, R. Rumely and ourselves. In particular our results, when combined with techniques of diophantine approximation, will allow us to compute the integrals in the generalized Mahler formula by averaging on periodic points. 1.
On the spectrum of the ZhangZagier height
 Biological Cybernetics
, 1997
"... Abstract. From recent work of Zhang and of Zagier, we know that their height H(α) is bounded away from 1 for every algebraic number α different from 0, 1, 1/2 ± √ −3/2. The study of the related spectrum is especially interesting, for it is linked to Lehmer’s problem and to a conjecture of Bogomolov ..."
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Cited by 4 (1 self)
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Abstract. From recent work of Zhang and of Zagier, we know that their height H(α) is bounded away from 1 for every algebraic number α different from 0, 1, 1/2 ± √ −3/2. The study of the related spectrum is especially interesting, for it is linked to Lehmer’s problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the socalled ZhangZagier inequality. To achieve this, we need some algebraic numbers of small height. So, in the third section, we describe an algorithm able to find them, and we give an algebraic number with height 1.2875274... discovered in this way. This search up to degree 64 suggests that the spectrum of H(α) mayhave a limit point less than 1.292. We prove this fact in the fourth part. 1.