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Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of ...

57N10

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  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

Single angle  The Chebotarev density theorem
Stevenhagen, Peter (Auteur de la Conférence) | CIRM (Editeur )

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

This talk presents some news on bilinear decompositions of the Möbius function. In particular, we will exhibit a family of such decompositions inherited from Motohashi's proof of the Hoheisel Theorem that leads to
$\sum_{n\leq X,(n,q)=1) }^{} \mu (n)e(na/q)\ll X\sqrt{q}/\varphi (q)$
for $q \leq X^{1/5}$ and any $a$ prime to $q$.

11N37 ; 11Y35 ; 11A25

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

Hyperbolically embedded subgroups have been defined by Dahmani-Guirardel-Osin and they provide a common perspective on (relatively) hyperbolic groups, mapping class groups, Out(F_n), CAT(0) groups and many others. I will sketch how to extend a quasi-cocycle on a hyperbolically embedded subgroup H to a quasi-cocycle on the ambient group G. Also, I will discuss how some of those extended quasi-cocycles (of dimension 2 and higher) "contain" the information that H is hyperbolically embedded in G. This is joint work with Roberto Frigerio and Maria Beatrice Pozzetti. Hyperbolically embedded subgroups have been defined by Dahmani-Guirardel-Osin and they provide a common perspective on (relatively) hyperbolic groups, mapping class groups, Out(F_n), CAT(0) groups and many others. I will sketch how to extend a quasi-cocycle on a hyperbolically embedded subgroup H to a quasi-cocycle on the ambient group G. Also, I will discuss how some of those extended quasi-cocycles (of dimension 2 and higher) "contain" the ...

20F65

Single angle  Pre-adic and adic spaces
Buzzard, Kevin (Auteur de la Conférence) | CIRM (Editeur )

Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ,g;x)\gg \frac{x}{\left ( \log \log x\right )^{1+\varepsilon }}$,
while we prove that $E(\varepsilon ,g;x)=o(x)$ as $x\rightarrow \infty$ for every multiplicative function $g$.
Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ...

11N37 ; 11K65 ; 11N60

This is a joint work with J. Cogdell and T.-L. Tsai. I will report on the progress made in proving the equality of Artin epsilon factors for exterior and symmetric square L-functions with those on the representation theoretic side through the local Langlands correspondence. The equality for L-functions has already been established by Henniart. I will show how the equality can be proved if one has the stability of these factors under highly ramified twists for supercuspidal representations. I will then discuss the stability question for supercuspidals by discussing how it can be deduced from a generalization of germ expansions of Jacquet and Ye from Bessel functions to certain partial Bessel functions. I will elaborate by explaining the stability in the case of GL(2) through general lemmas proved so far. This is a joint work with J. Cogdell and T.-L. Tsai. I will report on the progress made in proving the equality of Artin epsilon factors for exterior and symmetric square L-functions with those on the representation theoretic side through the local Langlands correspondence. The equality for L-functions has already been established by Henniart. I will show how the equality can be proved if one has the stability of these factors under highly ...

11F66 ; 11F70 ; 11F80 ; 22E50

Let $s(m)$ denote the number of distinct powers of 2 in the binary representation of $m$. Thus the Thue-Morse sequence is $(-1)^{s(m)}$ and
$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1)^{s(p)}$ depends on nontrivial bounds for $\left \| T_n \right \|_1$ and for $\left \| T_n \right \|_\infty $. We consider other norms of the $T_n$. For positive integers $k$ let
$M_k(n)=\int_{0}^{1}\left | T_n(x) \right |^{2k}dx$
We show that the sequence $M_k(n)$ satisfies a linear recurrence of order $k$. Moreover, we determine a $k\times k$ matrix whose characteristic polynomial determines this linear recurrence.
This is joint work with Mauduit and Rivat.
Let $s(m)$ denote the number of distinct powers of 2 in the binary representation of $m$. Thus the Thue-Morse sequence is $(-1)^{s(m)}$ and
$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1)^{s(p)}$ depends on nontrivial bounds for $\left \| T_n \right \|_1$ and for $\left \| T_n \right \|_\infty $. We consider oth...

11B83

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

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  Hilbert cubes in arithmetic sets
Elsholtz, Christian (Auteur de la Conférence) | CIRM (Editeur )

Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $ \left |A \right |,\left |B \right |,\left |C \right |\geq 2$). More generally, in recent work we showed (with A. Harper) for certain multiplicatively defined sets $S$, namely those which can be treated by sieves, or those with some equidistribution condition of Bombieri-Vinogradov type, that again there is no (nontrivial) ternary decomposition $P\sim A+B+C$. As this covers the case of smooth numbers, this settles a conjecture of A.Sárközy.
Joint work with Adam J. Harper.
Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $ \left |A \right |,\left |B \right |,\left |C \right |\geq 2$). More generally, in recent work we showed (with A. Harper) for certain ...

11-XX ; 05-XX

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include ...

11G20 ; 14G15 ; 14H52

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include ...

11G20 ; 14G15 ; 14H52

We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions.
These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography.
In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.
We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like.
This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include ...

11G20 ; 14G15 ; 14H52

In this final, fourth lecture the many effects on radial tokamak plasma transport caused by various physical processes are noted first: transients, collision- and microturbulence-induced transport, sources and sinks, and small three-dimensional (3-D) magnetic field perturbations. The main focus of this lecture is on the various effects of small 3-D fields on plasma transport which is a subject that has come of age over the past decade. Finally, the major themes of these CEMRACS 2014 lectures are summarized and a general framework for combining extended MHD, hybrid kinetic/fluid and transport models of tokamak plasma behavior into unified descriptions and numerical simulations that may be able to provide a "predictive capability" for ITER plasmas is presented. In this final, fourth lecture the many effects on radial tokamak plasma transport caused by various physical processes are noted first: transients, collision- and microturbulence-induced transport, sources and sinks, and small three-dimensional (3-D) magnetic field perturbations. The main focus of this lecture is on the various effects of small 3-D fields on plasma transport which is a subject that has come of age over the past decade. Finally, ...

76X05 ; 82C70

In this third lecture the ideal and extended magnetohydrodynamics (MHD) fluid moment descriptions of magnetized plasmas are discussed first. The ideal MHD equilibrium in a toroidally axisymmetric tokamak plasma is discussed next. Then, the collisional viscous force closure moments and their effects on the parallel Ohm's law and poloidal flows in the extended MHD model of tokamak plasmas are discussed. Finally, the species fluid moment equations are transformed to magnetic flux coordinates, averaged over a flux surface and used to obtain the tokamak plasma transport equations. These equations describe the transport of the plasma electron density, plasma toroidal angular momentum and pressure of the electron and ion species "radially" across the nested tokamak toroidal magnetic flux surfaces. In this third lecture the ideal and extended magnetohydrodynamics (MHD) fluid moment descriptions of magnetized plasmas are discussed first. The ideal MHD equilibrium in a toroidally axisymmetric tokamak plasma is discussed next. Then, the collisional viscous force closure moments and their effects on the parallel Ohm's law and poloidal flows in the extended MHD model of tokamak plasmas are discussed. Finally, the species fluid moment equations ...

76X05 ; 82C70

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