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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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Research talks

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations. Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a ...

11M26 ; 11M06 ; 11F66 ; 11T55 ; 11R44 ; 11M50

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Research talks

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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H

Research talks

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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Sato-Tate axioms Fité, Francesc | CIRM H

Single angle

H

Research talks

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 ; 14G10 ; 14K15

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H

Research talks

Moment sequences as a tool for identifying and classifying Sato-Tate distributions. Computing moment sequences of Sato-Tate groups, Weyl integration formulas, comparing moment statistics, distinguishing exceptional distributions with additional statistics.
Sato-Tate - Abelian surfaces - Abelian threefolds - hyperelliptic curves

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

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H

Research talks

Overview of the generalized Sato-Tate conjecture with lots of explicit examples. Preliminary discussion of L-polynomial distributions, Sato-Tate groups, and moment sequences. Presentation of the main results in genus 2.
Sato-Tate - Abelian surfaces - Abelian threefolds - hyperelliptic curves

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

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