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Connectivity of Markoff mod-p graphs - Fuchs, Elena (Auteur de la conférence) | CIRM H

Multi angle

The study of Markoff triples, solutions to $ x^{2} + y^{2} + z^{2} = 3xyz $, spans over many fields. In this talk, we discuss arithmetic of Markoff triples by considering Markoff mod-p graphs. We will delve into what is known about their connectivity, going into a recent result which is joint work with Eddy, Litman, Martin, and Tripeny.

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Billiards and number theory - McMullen, Curtis T. (Auteur de la conférence) | CIRM H

Multi angle

The dynamics of billiards in a regular polygon can be studied from a wide range of mathematical perspectives. In this talk we will discuss connections between periodic billiard paths and number theory, modular symbols, heights and Hilbert modular varieties. In particular we will describe a new form of arithmetic chaos that emerges from dynamics in the regular 12-gon.

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An introduction to Walnut - lecture 1 - Rampersad, Narad (Auteur de la conférence) | CIRM H

Multi angle

Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for participants to get some hands-on training on how to use Walnut.[-]
Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for ...[+]

68R15 ; 68Q45 ; 03F30

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An introduction to Walnut - lecture 2 - Rampersad, Narad (Auteur de la conférence) | CIRM H

Multi angle

Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for participants to get some hands-on training on how to use Walnut.[-]
Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for ...[+]

68R15 ; 68Q45 ; 03F30

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Functional equations and combinatorics - Di Vizio, Lucia (Auteur de la conférence) | CIRM H

Multi angle

Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the audience is interested.[-]
Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the ...[+]

12H05 ; 05A15 ; 11B68 ; 05A40 ; 33B15 ; 33C45 ; 39A10 ; 30D30

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In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will discuss how this generalizes (conjecturally) to triangulated motives and thus to motivic cohomology.[-]
In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will ...[+]

14F42

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Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco.[-]
Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the ...[+]

14G05 ; 11G35

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We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively.[-]
We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...[+]

11N05 ; 11B05

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Distributions of Frobenius of elliptic curves #3 - Jones, Nathan (Auteur de la conférence) | CIRM H

Single angle

In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in $GL_2$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results.

Various questions in number theory may be viewed in probabilistic terms. For instance, consider the prime number theorem, which states that, as $x\rightarrow \infty$ , one has
$\#\left \{ primes\, p\leq x \right \}\sim \frac{x}{\log x}$
This may be seen as saying that the heuristic “probability” that a number $p$ is prime is about $1/\log p$. This viewpoint immediately predicts the correct order of magnitude for the twin prime conjecture. Indeed, if $p$ and $p+2$ are seen as two randomly chosen numbers of size around $t$, then the probability that they are both prime should be about $1/(\log t)^2$, which predicts that
$\#\left \{ primes\, p\leq x : p+2\, is\, also\, prime \right \}\asymp \int_{2}^{x}\frac{1}{(\log t)^2}dt \sim \frac{x}{\log x}$
In this naive heuristic, the events “$p$  is prime” and “$p+2$ is prime” have been treated as independent, which they are not (for instance their reductions modulo 2 are certainly not independent). Using more careful probabilistic reasoning, one can correct this and arrive at the precise conjecture
$\#\left \{ primes\, p\leq x : p+2\, is\, also\, prime \right \} \sim C_{twin}\frac{x}{(\log x)^2}$,
where $C_{twin}$  is the constant of Hardy-Littlewood.
In these lectures, we will use probabilistic considerations to study statistics of data attached to elliptic curves. Specifically, fix an elliptic curve $E$  over $\mathbb{Q}$ of conductor $N_E$. For a prime $p$ of good reduction, theFrobenius trace $a_p(E)$ and Weil $p$-root $\pi _p(E)\in \mathbb{C}$ satisfy the relations
$\#E(\mathbb{F}_p)=p+1-a_p(E)$,
$X^2-a_p(E)X+p=(X-\pi _p(E))(X-\overline{ \pi _p(E)})$.
Because of their connection via the Birch and Swinnerton-Dyer conjecture to ranks of elliptic curves (amongother reasons), there is general interest in understanding the statistical variation of the numbers $a_p(E)$ and $\pi_p(E)$, as $p$ varies over primes of good reduction for E. In their 1976 monograph, Lang and Trotter considered the following two fundamental counting functions:
$\pi_{E,r}(x) :=\#\left \{ primes\: p\leq x:p \nmid N_E, a_p(E)=r \right \}$
$\pi_{E,K}(x) :=\#\left \{ primes\: p\leq x:p \nmid N_E, \mathbb{Q}(\pi_p(E))=K \right \}$,
where $ r \in \mathbb{Z}$ is a fixed integer, $K$ is a fixed imaginary quadratic field. We will discuss their probabilistic model, which incorporates both the Chebotarev theorem for the division fields of $E$ and the Sato-Tatedistribution, leading to the precise (conjectural) asymptotic formulas
(1) $\pi_{E,r}(x)\sim C_{E,r}\frac{\sqrt{x}}{\log x}$
$\pi_{E,K}(x)\sim C_{E,K}\frac{\sqrt{x}}{\log x}$,
with explicit constants$C_{E,r}\geq 0$ and $C_{E,K} > 0$. We will also discuss heuristics leading to the conjectureof Koblitz on the primality of $\#E( \mathbb{F}_p)$, and of Jones, which combines these with the model of Lang-Trotter for $\pi_{E,r}(x)$ in order to count amicable pairs and aliquot cycles for elliptic curves as introduced by Silvermanand Stange.
The above-mentioned conjectures are all open, although (in addition to the bounds mentioned in the previous section) there are various average results which give evidence of their validity. For instance, let $R\geq 1$ and $S\geq 1$be an arbitrary positive length andwidth, respectively, and define
$\mathcal{F}(R,S):= \{ E_{r,s}:(r,s)\in \mathbb{Z}^2,-16(4r^3+27s^2)\neq 0, \left | r \right |\leq R\: $ and $\left | s \right | \leq S \}$,
where $E_{r,s}$ denotes the curve with equation $y^2=x^3+rx=s$. The work of Fouvry and Murty $(r=0)$, and of David and Pappalardi $(r\neq 0)$, shows that, provided min $\left \{ R(x), S(x) \right \}\geq x^{1+\varepsilon }$, one has
(2) $\frac{1}{\left |\mathcal{F}(R(x),S(x)) \right |} \sum_{E\in \mathcal{F}(R(x),S(x))} \pi_{E,r}(x) \sim C_r \frac{\sqrt{x}}{\log x}$
where $C_r$ is a constant. We will survey this and other theorems on average, and then discuss the nature of the associated constants $C_{E,r},C_{E,K}$ etc. We will discuss the statistical variation of these constants as $E$ varies over all elliptic curves over $\mathbb{Q}$, and use this to confirm the consistency of (2) with (1), on the level of the constants

Keywords : Galois representation - elliptic curve - trace of Frobenius - Chebotarev density theorem - Sato-Tate conjecture - Lang-Trotter conjecture[-]
In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in $GL_2$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results.

Various ...[+]

11G05 ; 11R44

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Introduction to Sato-Tate distributions - Sutherland, Andrew (Auteur de la conférence) | CIRM H

Single angle

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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