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49
Small points on subvarieties of a torus
 Journal
, 2009
"... Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We the ..."
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Cited by 21 (3 self)
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Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees. As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor. 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
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 17 (4 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Factoring bivariate sparse (lacunary) polynomials
 J. Complexity
"... Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Mo ..."
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Cited by 13 (2 self)
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Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d.
Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze
, 2012
"... We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of proje ..."
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Cited by 10 (2 self)
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We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed heights of multiprojective varieties. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.
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.
Examples of torsion points on genus two curves
 Trans. of the AMS
"... Abstract. We describe a method that sometimes determines all the torsion points lying on a curve of genus two dened over a number eld and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples y2 = x5 + x, y2 = x5 + 5x3 + x, and y2 y = x5. ..."
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Cited by 9 (0 self)
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Abstract. We describe a method that sometimes determines all the torsion points lying on a curve of genus two dened over a number eld and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples y2 = x5 + x, y2 = x5 + 5x3 + x, and y2 y = x5.
An Arakelovtheoretic approach to naive heights on hyperelliptic Jacobians
 JENNIFER S. BALAKRISHNAN, AMNON BESSER, AND J. STEFFEN MÜLLER
"... ar ..."
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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.