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Documents  Shparlinski, Igor | enregistrements trouvés : 26

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Outreach;Mathematics Education and Popularization of Mathematics

Igor Shparlinski held the Jean Morlet Chair from February 2014 to August 2014. This chair was linked in parts to the thematic month on 'Arithmetics' which took part in February 2014 at CIRM. Igor Shparlinski has a career in Number theory and its applications to cryptography, with significant overlap with the research interests of the groups Dynamique Arithmétique, Combinatoire (DAC) and Arithmétique et Théorie de l'Information (ATI) in Marseille. The idea was to start the month with a week on 'Unlikely Intersections' followed by a workshop organized by members of the DAC research group. Weeks 3 and 4 were on 'Frobenius distributions' and were co-organized with the ATI group. The focus was to introduce and explore new directions of research around the proof of the Sato-Tate conjecture, its generalizations, and the related Lang-Trotter conjecture. Continuing the progression to the interactions of arithmetics with geometry, the thematic month closed with a week on the topic 'On the Conjectures of Lang and Volta'.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
Igor Shparlinski held the Jean Morlet Chair from February 2014 to August 2014. This chair was linked in parts to the thematic month on 'Arithmetics' which took part in February 2014 at CIRM. Igor Shparlinski has a career in Number theory and its applications to cryptography, with significant overlap with the research interests of the groups Dynamique Arithmétique, Combinatoire (DAC) and Arithmétique et Théorie de l'Information (ATI) in ...

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Research talks;Lie Theory and Generalizations;Number Theory

We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the characteristic function, the density, and the repartition function of this distribution in terms of higher transcendental functions, namely Legendre and Meijer functions. We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the cha...

11G05 ; 11G10 ; 14G10 ; 37C30

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- 62 p.
ISBN 978-2-85629-786-5

Localisation : Colloque 1er étage (PARI)

dynamique hyperbolique # théorie des nombres # cryptographie

37D40 ; 11T71

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- viii; 238 p.
ISBN 978-1-4704-1947-9

Contemporary mathematics , 0663

Localisation : Collection 1er étage

distribution de Frobenius # conjecture de Sato-Tate # conjecture de Lang-Trotter

11G05 ; 11G10 ; 11G20 ; 11G25 ; 11M50 ; 11N05 ; 11R44 ; 14G10 ; 14G25

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Localisation : Colloque 1er étage (SYDN)

Prolog # algorithme résultant # algèbre et informatique # calcul de 2-cocycle # calcul de groupe 2-classe quadratique # corps quadratique imaginaire # graphe planaire maximal # meilleure approximation # processus de sous-groupe à faible indice # suite d'élément de semi-groupe # théorie des nombres

05Cxx ; 11Rxx ; 11Yxx ; 20Exx ; 68Txx

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Research talks;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry;Number Theory

For a polynomial $f(x)$ over a field $L$, and an element $c \in L$, I will discuss the size of the intersection of the orbit $\lbrace f(c),f(f (c)),...\rbrace$ with a prescribed subfield of $L$. I will also discuss the size of the intersection of orbits of two distinct polynomials, and generalizations of these questions to more general maps between varieties.
polynomial decomposition - classification of finite simple groups - Bombieri-Lang conjecture - orbit - dynamical system - unlikely intersections
For a polynomial $f(x)$ over a field $L$, and an element $c \in L$, I will discuss the size of the intersection of the orbit $\lbrace f(c),f(f (c)),...\rbrace$ with a prescribed subfield of $L$. I will also discuss the size of the intersection of orbits of two distinct polynomials, and generalizations of these questions to more general maps between varieties.
polynomial decomposition - classification of finite simple groups - Bombieri-Lang ...

11C08 ; 14Gxx ; 37F10

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- 318 p.
ISBN 978-0-8218-3387-2

Mathematical surveys and monographs , 0104

Localisation : Collection 1er étage

théorie des nombres # suite définie par récurrence # recurrence # suite automatique # courbe elliptique # distribution modulo 1 # nombre pseudo-alléatoire # cryptographie # somme exponentielle # automate cellulaire

11B37 ; 11B85 ; 11G05 ; 11J71 ; 11K45 ; 11T23 ; 37B15 ; 94A60

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Research talks;Number Theory

This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the second talk, we present the Sato-Tate axiomatic, which leads us to some Lie group theoretic classification results. The last part of the talk is devoted to illustrate the methods involved in the proof of this kind of results by considering a concrete example. In the third and final talk, we present Banaszak and Kedlaya's algebraic version of the Sato-Tate conjecture, we describe the notion of Galois type of an abelian variety, and we establish the dictionary between Galois types and Sato-Tate groups of abelian surfaces defined over number fields.
generalized Sato-Tate conjecture - Sato-Tate group - equidistribution - Sato-Tate axioms - Galois type - Abelian surfaces - endomorphism algebra - Frobenius distributions
This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the ...

11M50 ; 11G10 ; 11G20 ; 14G10 ; 14K15

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Research talks;Number Theory

This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the second talk, we present the Sato-Tate axiomatic, which leads us to some Lie group theoretic classification results. The last part of the talk is devoted to illustrate the methods involved in the proof of this kind of results by considering a concrete example. In the third and final talk, we present Banaszak and Kedlaya's algebraic version of the Sato-Tate conjecture, we describe the notion of Galois type of an abelian variety, and we establish the dictionary between Galois types and Sato-Tate groups of abelian surfaces defined over number fields.
generalized Sato-Tate conjecture - Sato-Tate group - equidistribution - Sato-Tate axioms - Galois type - Abelian surfaces - endomorphism algebra - Frobenius distributions
This series of three talks is the first part of an introductory course on the generalized Sato-Tate conjecture, made in collaboration with Andrew V. Sutherland at the Winter School "Frobenius distributions on curves", celebrated in Luminy in February 2014. In the first talk, some general background following Serre's works is introduced: equidistribution and its connexion to L-functions, the Sato-Tate group and the Sato-Tate conjecture. In the ...

11M50 ; 11G10 ; 11G20 ; 14G10 ; 14K15

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Research talks;Number Theory

We explain Chebotarev’s theorem, which is The Fundamental Tool in proving whatever densities we have for sets of prime numbers, try to understand what makes it hard in the case of ifinite extensions, and see why such extensions arise in the case of primitive root problems.

11R45

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