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Documents  11G05 | enregistrements trouvés : 18

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Post-edited  The symplectic type of congruences between elliptic curves Cremona, John (Auteur de la Conférence) | CIRM (Editeur )

In this talk I will describe a systematic investigation into congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$. For each such curve $E$ and prime $p$ the $p$-torsion $E[p]$ of $E$, is a 2-dimensional vector space over $\mathbb{F}_{p}$ which carries a Galois action of the absolute Galois group $G_{\mathbb{Q}}$. The structure of this $G_{\mathbb{Q}}$-module is very well understood, thanks to the work of J.-P. Serre and others. When we say the two curves $E$ and $E'$ are ”congruent” we mean that $E[p]$ and $E'[p]$ are isomorphic as $G_{\mathbb{Q}}$-modules. While such congruences are known to exist for all primes up to 17, the Frey-Mazur conjecture states that p is bounded: more precisely, that there exists $B$ > 0 such that if $p > B$ and $E[p]$ and $E'[p]$ are isomorphic then $E$ and $E'$ are isogenous. We report on work toward establishing such a bound for the elliptic curves in the LMFDB database. Secondly, we describe methods for determining whether or not a given isomorphism between $E[p]$ and $E'[p]$ is symplectic (preserves the Weil pairing) or antisymplectic, and report on the results of applying these methods to the curves in the database.
This is joint work with Nuno Freitas (Warwick).
In this talk I will describe a systematic investigation into congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$. For each such curve $E$ and prime $p$ the $p$-torsion $E[p]$ of $E$, is a 2-dimensional vector space over $\mathbb{F}_{p}$ which carries a Galois action of the absolute Galois group $G_{\mathbb{Q}}$. The structure of this $G_{\mathbb{Q}}$-module is very well understood, thanks to the work of ...

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Post-edited  Stable models for modular curves in prime level Parent, Pierre (Auteur de la Conférence) | CIRM (Editeur )

We describe stable models for modular curves associated with all maximal subgroups in prime level, including in particular the new case of non-split Cartan curves.
Joint work with Bas Edixhoven.

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Post-edited  Integral points on Markoff type cubic surfaces and dynamics Sarnak, Peter (Auteur de la Conférence) | CIRM (Editeur )

Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space.
Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the ...

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Post-edited  Formulas for the limiting distribution of traces of Frobenius Lachaud, Gilles (Auteur de la Conférence) | CIRM (Editeur )

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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Post-edited  Heuristics for boundedness of ranks of elliptic curves Poonen, Bjorn (Auteur de la Conférence) | CIRM (Editeur )

We present heuristics that suggest that there is a uniform bound on the rank of $E(\mathbb{Q})$ as $E$ varies over all elliptic curves over $\mathbb{Q}$. This is joint work with Jennifer Park, John Voight, and Melanie Matchett Wood.

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Multi angle  Monogenic cubic fields and local obstructions Shnidman, Ari (Auteur de la Conférence) | CIRM (Editeur )

A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that for any degree d > 2, the proportion of degree d number fields which are monogenic is 0. There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I’ll discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are still not monogenic. This uses new results on ranks of Selmer groups of elliptic curves in twist families.
A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that for any degree d > 2, the proportion of degree d number fields which are monogenic is 0. There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I’ll discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are ...

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Multi angle  Isolated points on modular curves Viray, Bianca (Auteur de la Conférence) | CIRM (Editeur )

Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that al but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^{1}$ or positive rank abelian varieties, we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_{1}(n)$ push down to isolated points on aj only on the $j$-invariant of the isolated point.
This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.
Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that al but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^{1}$ or positive rank abelian varieties, we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_{1}(n)$ push down ...

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Multi angle  Height pairings, torsion points, and dynamics Krieger, Holly (Auteur de la Conférence) | CIRM (Editeur )

We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of elliptic curves.
We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of ...

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Multi angle  Shimura curves and bounds for the $abc$ conjecture Pasten, Hector (Auteur de la Conférence) | CIRM (Editeur )

I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.
I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the ...

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Multi angle  Recent progress in the classification of torsion subgroups of elliptic curves Lozano-Robledo, Alvaro (Auteur de la Conférence) | CIRM (Editeur )

This talk will be a survey of recent results and methods used in the classification of torsion subgroups of elliptic curves over finite and infinite extensions of the rationals, and over function fields.

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Multi angle  Computing the image of Galois representations attached to elliptic curves Sutherland, Andrew (Auteur de la Conférence) | CIRM (Editeur )

Let $E$ be an elliptic curve over a number field $K$. For each integer $n > 1$ the action of the absolute Galois group $G_K := Gal(\overline{K}/K)$ on the $n$-torsion subgroup $E [n]$ induces a Galois representation $\rho_{E,n}:G_K \rightarrow$ Aut$(E[n]) \backsimeq GL_2(\mathbb{Z} /n\mathbb{Z})$. The representations $\rho_{E,n}$ form a compatible system, and after taking inverse limits one obtains an adelic representation $\rho_E:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$. If $E/K$ does not have $CM$, then Serre’s open image theorem implies that the image of $\rho_E$ has finite index in $GL_2(\hat{\mathbb{Z}})$; in particular, $\rho_{E,\ell}$ is surjective for all but finitely many primes $\ell$.
I will present an algorithm that, given an elliptic curve $E/K$ without $CM$, determines the image of $\rho_{E,\ell}$ in $GL_2(\mathbb{Z} /\ell\mathbb{Z})$ up to local conjugacy for every prime $\ell$ for which $\rho_{E,\ell}$ is non-surjective. Assuming the generalized Riemann hypothesis, the algorithm runs in time that is polynomial in the bit-size of the coefficients of an integral Weierstrass model for $E$. I will then describe a probabilistic algorithm that uses this information to compute the index of $\rho_E$ in $GL_2(\hat{\mathbb{Z}})$.
Let $E$ be an elliptic curve over a number field $K$. For each integer $n > 1$ the action of the absolute Galois group $G_K := Gal(\overline{K}/K)$ on the $n$-torsion subgroup $E [n]$ induces a Galois representation $\rho_{E,n}:G_K \rightarrow$ Aut$(E[n]) \backsimeq GL_2(\mathbb{Z} /n\mathbb{Z})$. The representations $\rho_{E,n}$ form a compatible system, and after taking inverse limits one obtains an adelic representation $\rho_E:G_K \... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Multi angle How to prove that Galois groups are large Serre, Jean-Pierre (Auteur de la Conférence) | CIRM (Editeur ) The Galois groups of the title are those which are associated with elliptic curves over number fields; I shall explain the methods which were introduced in the 1960's in order to prove that they are large, and the questions about them which are still open fifty years later. Galois - elliptic - l-adic - Tate - proofs Déposez votre fichier ici pour le déplacer vers cet enregistrement. Single angle Distributions of Frobenius of elliptic curves #4 Jones, Nathan (Auteur de la Conférence) | CIRM (Editeur ) In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in$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$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results. Various ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Single angle Distributions of Frobenius of elliptic curves #6 Jones, Nathan (Auteur de la Conférence) | CIRM (Editeur ) In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in$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$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results. Various ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Single angle Distributions of Frobenius of elliptic curves #1 David, Chantal (Auteur de la Conférence) | CIRM (Editeur ) In all the following, let an elliptic curve$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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Single angle  Distributions of Frobenius of elliptic curves #2 David, Chantal (Auteur de la Conférence) | CIRM (Editeur )

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$(K_\ell/\mathbb{Q})\subseteq GL_2... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Single angle Distributions of Frobenius of elliptic curves #5 David, Chantal (Auteur de la Conférence) | CIRM (Editeur ) In all the following, let an elliptic curve$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...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Single angle  Distributions of Frobenius of elliptic curves #3 Jones, Nathan (Auteur de la Conférence) | CIRM (Editeur )

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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