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Post-edited  Avoiding Jacobians
Masser, David (Auteur de la Conférence) | CIRM (Editeur )

It is classical that, for example, there is a simple abelian variety of dimension $4$ which is not the jacobian of any curve of genus $4$, and it is not hard to see that there is one defined over the field of all algebraic numbers $\overline{\bf Q}$. In $2012$ Chai and Oort asked if there is a simple abelian fourfold, defined over $\overline{\bf Q}$, which is not even isogenous to any jacobian. In the same year Tsimerman answered ''yes''. Recently Zannier and I have done this over the rationals $\bf Q$, and with ''yes, almost all''. In my talk I will explain ''almost all'' the concepts involved.
It is classical that, for example, there is a simple abelian variety of dimension $4$ which is not the jacobian of any curve of genus $4$, and it is not hard to see that there is one defined over the field of all algebraic numbers $\overline{\bf Q}$. In $2012$ Chai and Oort asked if there is a simple abelian fourfold, defined over $\overline{\bf Q}$, which is not even isogenous to any jacobian. In the same year Tsimerman answered ''yes''. ...

14H40 ; 14K02 ; 14K15 ; 11G10

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Multi angle  Arithmetic of rank one local systems
Esnault, Hélène (Auteur de la Conférence) | CIRM (Editeur )

Joint with Moritz Kerz. We study arithmetic subvarieties of the character variety of normal complex varieties defined over a field of finite type.

14D20 ; 14F05 ; 14F10 ; 14F30 ; 14K15

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I will explain how previous (conditional) minimal modularity lifting results (in the presence of torsion) may be adapted to the non-minimal case in the context of imaginary quadratic fields. This is joint work with David Geraghty.

11F33 ; 11F80 ; 14K15

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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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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

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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

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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

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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

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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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