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Documents  Zannier, Umberto | enregistrements trouvés : 14

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Research talks;Number Theory

We shall discuss joint work with Robert and Tenenbaum on a proposed refinement of the well known abc conjecture.

11N25 ; 11Dxx ; 11N56

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Research talks;Combinatorics;Number Theory

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler’s method provides one of the few known instances where it essentially holds true. After the works of Nishioka, and more recently of Philippon, Faverjon and the speaker, the theory of Mahler functions in one variable is now rather well understood. In contrast, and despite the contributions of Mahler, Loxton and van der Poorten, Kubota, Masser, and Nishioka among others, the theory of Mahler functions in several variables remains much less developed. In this talk, I will discuss recent progresses concerning the case of regular singular systems, as well as possible applications of this theory. This is a joint work with Colin Faverjon. Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler’s method provides ...

11J81 ; 11J85 ; 11B85

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- 57p.
ISBN 978-88-7642-212-6

Localisation : Colloque 1er étage (PISA)

brochure commémorative # dynamique algébrique # valeur propre statistique # variété abélienne # algèbre de Hopf # équation diophantienne

00B25 ; 00B30 ; 00B10

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- xii, 98 p.
ISBN 978-88-7642-344-4

Colloquia , 0002

Localisation : Colloque 1er étage (PISA)

00B25 ; 60-06

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- xvii; 390 p.
ISBN 978-88-7642-206-5

CRM Series , 0004

Localisation : Colloque 1er étage (PISA)

Géométrie diophantienne

11-06 ; 14-06 ; 00B25 ; 11Gxx ; 14Gxx

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Research talks;Number Theory

The assertions made by L. J. Mordell in his paper in Acta Mathematica 44(1952) are discussed. Mordell had been to a certain extent anticipated by E. Jacobsthal (1939).
backward induction - congruence - equation - non-zero coefficients - polynomials

11D09 ; 11D25 ; 11D41

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Research talks;Number Theory

A polynomial $D(t)$ is called Pellian if the ring generated over $C[t]$ by its square root has non constant units. By work of Masser and Zannier on the relative Manin-Mumford conjecture for jacobians, separable sextic polynomials are usually not Pellian. The same applies in the non-separable case, though some exceptional families occur, in relation to Ribet sections on generalized jacobians.

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Research talks;Algebraic and Complex Geometry

An interpolation estimate is a sufficient condition for the evaluation map to be surjective; it is dual to a multiplicity estimate, which deals with injectivity. Masser's first interpolation estimate on commutative algebraic groups can be generalized, and made essentially as precise as the best known multiplicity estimates in this setting. As an application, we prove a result that connects interpolation and multiplicity estimates.
This is a joint work with M. Nakamaye.
An interpolation estimate is a sufficient condition for the evaluation map to be surjective; it is dual to a multiplicity estimate, which deals with injectivity. Masser's first interpolation estimate on commutative algebraic groups can be generalized, and made essentially as precise as the best known multiplicity estimates in this setting. As an application, we prove a result that connects interpolation and multiplicity estimates.
This is a ...

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Research talks;Number Theory

As is well known, simultaneous rational approximations to the values of smooth functions of real variables involve counting and/or understanding the distribution of rational points lying near the manifold parameterised by these functions. I will discuss recent results in this area regarding lower bounds for the Hausdorff dimension of $\tau$-approximable values, where $\tau\geq \geq 1/n$ is the exponent of approximations. In particular, I will describe a very recent development for non-degenerate maps as well as a recently introduced simple technique based on the so-called Mass Transference Principle that surprisingly requires no conditions on the functions except them being $C^2$. As is well known, simultaneous rational approximations to the values of smooth functions of real variables involve counting and/or understanding the distribution of rational points lying near the manifold parameterised by these functions. I will discuss recent results in this area regarding lower bounds for the Hausdorff dimension of $\tau$-approximable values, where $\tau\geq \geq 1/n$ is the exponent of approximations. In particular, I will ...

11J13 ; 11J83 ; 11K60

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Research talks;Number Theory

It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagranges theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known until now. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the $p$-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the ‘real’ value of the $p$-adic continued fraction. It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagranges theorem holds, and ...

11J70 ; 11D88 ; 11Y16

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Research talks;Algebraic and Complex Geometry;Number Theory

We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of elliptic curves. We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of ...

14G05 ; 11G50 ; 11G05

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Research talks;Number Theory

In a recent collaboration with Pascal Autissier and Marc Hindry, we prove that up to isomorphisms, there are at most finitely many elliptic curves defined over a fixed number field, with Mordell-Weil rank and regulator bounded from above, and rank at least 4.

11G50 ; 14G40

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- xi; 160 p.
ISBN 978-0-691-15371-1

Annals of mathematics studies , 0181

Localisation : Ouvrage RdC (ZANN)

théorie des intersections # variétés algébriques # géométrie algébrique # point rationnel # surface élliptique # variétés de Shimura # conjecture de Zilber # conjecture de Manin-Mumford # problème de Masser # conjecture d'Andre-Oort

14-02 ; 14C17 ; 14G05 ; 14G40 ; 14J27 ; 14G35 ; 11G18

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- x; 198 p.
ISBN 978-1-108-42494-3

Cambridge tracts in mathematics , 0212

Localisation : Collection 1er étage

analyse diophantienne # théorie des nombres # nombre transcendant

11-02 ; 11J13 ; 11J20 ; 11J68 ; 11J81 ; 11J87

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