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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

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

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

Single angle  Moment sequences of Sato-Tate groups
Sutherland, Andrew (Auteur de la Conférence) | CIRM (Editeur )

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

Single angle  Computing Sato-Tate statistics
Sutherland, Andrew (Auteur de la Conférence) | CIRM (Editeur )

Survey of methods for computing zeta functions of low genus curves, including generic group algorithms, p-adic cohomology, CRT-based methods (Schoof-Pila), and recent average polynomial-time algorithms.
Sato-Tate - Abelian surfaces - Abelian threefolds - hyperelliptic curves

11Y16 ; 11G10 ; 11G20 ; 14G10 ; 14K15

Single angle  The generalized Sato-Tate conjecture
Fité, Francesc (Auteur de la Conférence) | CIRM (Editeur )

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

Single angle  Sato-Tate axioms
Fité, Francesc (Auteur de la Conférence) | CIRM (Editeur )

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

Single angle  The Galois type of an Abelian surface
Fité, Francesc (Auteur de la Conférence) | CIRM (Editeur )

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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