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16
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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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.
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.
The intersection of a curve with a union of translated codimension 2 subgroups in a power of an elliptic curve
 Algebra and Number Theory
"... Abstract. Let E be an elliptic curve. Consider an irreducible algebraic curve C embedded in E g. The curve is transverse if it is not contained in any translate of a proper algebraic subgroup of E g. Furthermore C is weaktransverse if it is not contained in any proper algebraic subgroup. Suppose th ..."
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Abstract. Let E be an elliptic curve. Consider an irreducible algebraic curve C embedded in E g. The curve is transverse if it is not contained in any translate of a proper algebraic subgroup of E g. Furthermore C is weaktransverse if it is not contained in any proper algebraic subgroup. Suppose that both E and C are defined over the algebraic numbers. We prove that the algebraic points of a transverse curve C which are close to the union of all algebraic subgroups of E g of codimension 2 translated by points in a subgroup Γ of E g of finite rank are a set of bounded height. The notion of close is defined using a height function. If Γ is trivial, it is sufficient to suppose that C is weaktransverse. Then, we introduce a method to determine the finiteness of these sets. From a conjectural lower bound for the normalised height of a transverse curve C, we deduce that the above sets are finite. At present, such a lower bound exists for g≤3. Our results are optimal, for what concerns the codimension of the algebraic
SMALL POINTS ON RATIONAL SUBVARIETIES OF TORI.
, 2009
"... In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. ..."
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Cited by 2 (0 self)
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In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. Lehmer (see [Leh 1933])
Lower bounds for the normalized height and nondense subsets of subvarieties in an abelian variety
 Journal of Number Theory
"... Abstract. This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defin ..."
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Abstract. This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weaktransverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. For V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank are non Zariskidense in V. If Γ has rank zero, it is sufficient to assume that V is weaktransverse. The notion of closeness is defined using a height function. 1. introduction All varieties in this article are defined over Q. Denote by A a abelian variety of
Geometric lower bounds for the normalized height of hypersurfaces
, 2006
"... We are here concerned in the Bogomolov’s problem for the hypersurfaces; we give a geometric lower bound for the height of a hypersurface of G n m (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of G n m. This is an analogue of ..."
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We are here concerned in the Bogomolov’s problem for the hypersurfaces; we give a geometric lower bound for the height of a hypersurface of G n m (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of G n m. This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of nontorsion hypersurfaces defined and irreducible over the rationals.
Bogomolov on tori revisited
"... Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang), ..."
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Cited by 1 (1 self)
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Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang),
[Ominimalité et certaines intersections atypiques] OMINIMALITY AND CERTAIN ATYPICAL INTERSECTIONS
"... Abstract. We show that the strategy of point counting in ominimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of ManinMumford and AndréOort. We verify the socalled ZilberPink Conjecture in a product of modular curves on assuming a lowe ..."
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Abstract. We show that the strategy of point counting in ominimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of ManinMumford and AndréOort. We verify the socalled ZilberPink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular AxSchanuel Conjecture. In the context of abelian varieties we obtain the ZilberPink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results. On démontre que la stratégie de comptage dans des structures ominimales est suffisante pour traiter plusieurs problèmes qui vont audelà des conjectures de ManinMumford et AndréOort. On vérifie la conjecture de ZilberPink pour un produit de courbes modulaires en supposant une minoration assez forte pour la taille de l'orbite de Galois et en supposant une version modulaire du théorème de AxSchanuel. Dans le cas des variétés abéliennes on démontre la conjecture de ZilberPink pour les courbes si tous les objets sont définis sur un corps de nombres. Pour les sousvariétés de dimension supérieure on obtient quelques résultats plus faibles et quelques résultats conditionnels.