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# Documents  Ritzenthaler, Christophe | enregistrements trouvés : 28

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## Formulas for the limiting distribution of traces of Frobenius Lachaud, Gilles | CIRM H

Post-edited

Lie Theory and Generalizations;Number Theory

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...

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## Computing classical modular forms as orthogonal modular forms Voight, John | CIRM H

Post-edited

Algebraic and Complex Geometry;Number Theory

Birch gave an extremely efficient algorithm to compute a certain subspace of classical modular forms using the Hecke action on classes of ternary quadratic forms. We extend this method to compute all forms of non-square level using the spinor norm, and we exhibit an implementation that is very fast in practice. This is joint work with Jeffery Hein and Gonzalo Tornaria.

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## Arithmetic, geometry, cryptography and coding theoryinternational conferenceMarseille # november 5-9, 2007 Lachaud, Gilles ; Ritzenthaler, Christophe ; Tsfasman, Michael A. | American Mathematical Society 2009

Congrès

- vii; 206 p.
ISBN 978-0-8218-4716-9

Contemporary mathematics , 0487

Localisation : Collection 1er étage

théorie des nombres # géométrie algébrique # cryptographie # théorie des codes correcteurs d'erreurs

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## Arithmetic, geometry, cryptography and coding theory:13th conference arithmetic, geometry, cryptography and coding theory,CIRM, Marseille, France#March 14-18, 2011:Geocrypt 2011, Bastia, France#June 19-24, 2011 Aubry, Yves ; Ritzenthaler, Christophe ; Zykin, Alexey | American Mathematical Society 2012

Congrès

- vii-183 p.
ISBN 978-0-8218-7572-8

Contemporary mathematics , 0574

Localisation : Collection 1er étage

arithmétique # géométrie # cryptographie # codage # variétés abéliennes # théorie de la dimension

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## Arithmetic geometry: computation and applications.16th international conference on arithmetic, geometry, cryptography, and coding theoryMarseille # June 19-23, 2017 Aubry, Yves ; Howe, Everett W. ; Ritzenthaler, Christophe | American Mathematical Society 2019

Congrès

- vi; 175 p.
ISBN 978-1-4704-4212-5

Contemporary mathematics , 0722

Localisation : Collection 1er étage

théorie des nombres # théorie des codes # cryptographie # géométrie arithmétique # cohomologie de De Rham # fonction zeta

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## Les codes correcteurs Ritzenthaler, Christophe | CIRM H

Multi angle

Special events;Hommage à Claude Shannon;Computer Science;Mathematics Education and Popularization of Mathematics;History of Mathematics

A l'occasion du centenaire de la naissance de Claude Shannon, la SMF, la SMAI et le CIRM organisent, à l'issue de la conférence SIGMA, une après-midi d'exposés grand public autour de l'oeuvre scientifique de Claude Shannon, de la théorie de l'information et de ses applications.

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## Brauer-Siegel theorem and analogues for varieties over global fields Hindry, Marc | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some of the fundamental problems like the existence or non-existence of Siegel zeroes remains
unsolved, several generalisations and analog have been developed: Tsfasman-Vladuts, Kunyavskii-Tsfasman, Lebacque-Zykin, Hindry-Pacheco and lately Griffon. These analogues deal with number fields for which the limit is different from 1 or with elliptic curves and abelian varieties either for a fixed variety and varying field or over a fixed field with a family of varieties.
The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some ...

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## Maps between curves and diophantine obstructions Voloch, José Felipe | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to this situation and present some results and conjectures that arise. Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to ...

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## Local densities compute isogeny classes Achter, Jeffrey | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it's false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
I'll explain a transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, the new perspective allows a natural generalization to higher-dimensional abelian varieties. This is joint work with Julia Gordon and S. Ali Altug.
Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it's false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
I'll explain a transparent proof of this formula; it turns out that this product actually computes an adelic ...

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## Rational points on curves over finite fields Serre, Jean-Pierre ; Howe, Everett ; Oesterlé, Joseph ; Ritzenthaler, Christophe ; Bassa, Alp ; Lorenzo Garcia, Elisa ; Schoof, René | Société Mathématique de France 2020

Ouvrage

- x; 187 p.
ISBN 978-2-85629-923-4

Documents mathématiques , 0018

Localisation : Collection 1er étage

champ fini # courbe # point rationnel # théorie des corps de classes # résultat asymptotique # genre 2 # genre 3 # limite # formule explicite

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## Distributions of Frobenius of elliptic curves #3 Jones, Nathan | CIRM H

Single angle

Number Theory

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 ...

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## Introduction to Sato-Tate distributions Sutherland, Andrew | 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

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## Moment sequences of Sato-Tate groups Sutherland, Andrew | CIRM H

Single angle

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

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## Computing Sato-Tate statistics Sutherland, Andrew | CIRM H

Single angle

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

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## Group structures of elliptic curves #1 Shparlinski, Igor | CIRM H

Single angle

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.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
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 ...

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## Group structures of elliptic curves #2 Shparlinski, Igor | CIRM H

Single angle

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.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
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 ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Group structures of elliptic curves #3 Shparlinski, Igor | CIRM H

Single angle

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.
CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
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 ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Distributions of Frobenius of elliptic curves #4 Jones, Nathan | CIRM H

Single angle

Number Theory

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 ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Distributions of Frobenius of elliptic curves #6 Jones, Nathan | CIRM H

Single angle

Number Theory

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.

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## Distributions of Frobenius of elliptic curves #1 David, Chantal | CIRM H

Single angle

Number Theory

In all the following, let an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication. For every prime $\ell$, let $E[\ell]= E[\ell](\overline{\mathbb{Q}})$ be the group of $\ell$-torsion points of $E$, and let $K_\ell$ be the field extension obtained from $\mathbb{Q}$ by adding the coordinates of the $\ell$-torsion points of $E$. This is a Galois extension of$\mathbb{Q}$ , andGal$(K_\ell/\mathbb{Q})\subseteq GL_2(\mathbb{Z}/\ell\mathbb{Z})$.
Using the Chebotarev density theorem for the extensions $K_\ell/\mathbb{Q}$ associated to a given curve $E$, we can study various sequences associated to the reductions of a global curve $E/(\mathbb{Q}$, as the sequences
$\left \{\#E(\mathbb{F}_p)=p+1-a_p(E)\right \}_{p\: primes}, or \left \{ a_p(E)=r \right \}_{p\: primes}$
for some fixed value $r\in \mathbb{Z}$. For example, if $\pi_{E,r}(x)= \#\left \{ p\leq x : a_p(E)=r \right \}$,
then it was shown by Serre and K. Murty, R. Murty and Saradha that under the GRH,
$\pi_{E,r}(x)\ll x^{4/5} log^{-1/5}x$, for all $r\in \mathbb{Z}$, and $\pi_{E,0}(x)\ll x^{3/4}$.
There are also some weaker bounds without the GRH. Some other sequences may also be treated by apply-ing the Chebotarev density theorem to other extensions of $\mathbb{Q}$ as the ones coming from the “mixed Galois representations” associated to $E[\ell]$ and a given quadratic field $K$ which can be used to get upper bounds onthe number of primes $p$ such that End $(E/\mathbb{F}_p)\bigotimes \mathbb{Q}$ is isomorphic to a given quadratic imaginary field $K$ .
We will also explain how the densities obtained from the Cheboratev density theorem can be used togetherwith sieve techniques. For a first application, we consider a conjecture of Koblitz which predicts that
$\pi_{E}^{twin}(x):=\#\left \{ p\leq x : p+1-a_p(E)\, is\, prime \right \}\sim C_{E}^{twin}\frac{x}{log^2x}$
This is analogue to the classical twin prime conjecture, and the constant $C_{E}^{twin}$ can be explicitly writtenas an Euler product like the twin prime constant. We explain how classical sieve techniques can be usedto show that under the GRH, there are at least 2.778 $C_{E}^{twin}x/log^2x$ primes $p$ such that $p+1-a_p(E)^2$ has at most 8 prime factors, counted with multiplicity. We also explain some possible generalisation of Koblitz conjectures which could be treated by similar techniques given some explicitversions (i.e. with explicit error terms) of density theorems existing in the literature.
Other examples of sieving using the Chebotarev density theorem in the context of elliptic curves are thegeneralisations of Hooley’s proof of the Artin’s conjecture on primitive roots (again under the GRH).Using a similar techniques, but replacing the cyclotomic fields by the $\ell$-division fields $K_\ell$ of a given elliptic curve $E/\mathbb{Q}$, Serre showed that there is a positive proportion of primes $p$ such that the group $E(\mathbb{F}_p)$ is cyclic (when $E$ does not have a rational 2-torsion point). This was generalised by Cojocaru and Duke, and is also related to counting square-free elements of the sequence $a_p(E)^2-4p$,,which still resists a proof with the same techniques (without assuming results stronger than the GRH).
Finally, we also discuss some new distribution questions related to elliptic curves that are very similar to the questions that could be attacked with the Chebotarev density theorem, but are still completely open(for example, no non-trivial upper bounds exists). The first question was first considered by Silverman and Stange who defined an amicable pair of an elliptic curve $E/\mathbb{Q}$ to be a pair of primes $(p,q)$ such that
$p+1-a_p(E)=q$, and $q+1-a_q(E)=p$.
They predicted that the number of such pairs should be about $\sqrt{x}/log^2x$ for elliptic curves without complex multiplication. A precise conjecture with an explicit asymptotic was made by Jones, who also provided numerical evidence for his conjecture. Among the few results existing in the literature for thisquestion is the work of Parks who gave an upper bound of the correct order of magnitude for the average number (averaging over all elliptic curves) of amicable pairs (and aliquot cycles which are cycles of length $L$). But a non-trivial upper bound for a single elliptic curve is still not known.
Another completely open question is related to “champion primes”, which are primes $p$ such that $\#E(\mathbb{F}_p)$ is maximal, i.e. $a_p(E)=-[2\sqrt{p}]$. (This terminology was used for the first time by Hedetniemi, James andXue). In some work in progress with Wu, we make a conjecture and give some evidence for the number of champion primes associated to a given elliptic curve using the Sato-Tate conjecture (for verysmall intervals depending on $p$ i.e. in a range where the conjecture is still open). Again, this question iscompletely open, and there are no known non-trivial upper bound. There is also no numerical evidence for this question, and it would be nice to have some, possibly for more general “champion primes”, for examplelooking at $a_p(E)$ in a small interval of length $p^\varepsilon$ around $-[2\sqrt{p}]$.
In all the following, let an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication. For every prime $\ell$, let $E[\ell]= E[\ell](\overline{\mathbb{Q}})$ be the group of $\ell$-torsion points of $E$, and let $K_\ell$ be the field extension obtained from $\mathbb{Q}$ by adding the coordinates of the $\ell$-torsion points of $E$. This is a Galois extension of$\mathbb{Q}$ , andGal\$(K_\ell/\mathbb{Q})\subseteq GL_2...

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