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

Single angle

2y

Research talks

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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Introduction to Sato-Tate distributions Sutherland, Andrew | CIRM H

Single angle

H

Research talks

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

Single angle

H

Research talks

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

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

Single angle

H

Research talks

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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

Single angle

H

Research talks

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

leqslant 6$, we improve the preparation of$s$, so that its norm in the number field is less than$Q$. This improves its smoothness property. Assume that we want to compute the discrete logarithm of$s$in the larger subgroup of prime order$\ell$of$GF(p^n)$, with$\ell|\Phi_np$. We decompose$s$in$\epsilon \cdot s'$with$\epsilon$in a subfield or in a subgroup of order prime to$\ell$and$s?$with reduced coefficient size. We still have$log_g s = log_g s'$mod$\ell$. We use a tower representation of$GF(p^n)$with subfields for our purpose. We reduce the norm of$s \in \mathbb{F}_{p4}$from$O(p^{11/2})$to$O(p^{7/2}), s \in GF(p^3)$from$O(p^6)$to$O(p^2)$and$s \in \mathbb{F}_{p2}$from$O(p^4)$to$O(p)$. This does not change the asymptotic complexity of this last step but this improves a lot its running time for small$n$. This talk will focus on the last step of the number field sive algorithm used to compute discrete logarithms in finite fields. We consider here non-prime finite fields of very small extension degree:$1 \le n \le 6$. These cases are interesting in pairing-based cryptography: the pairing output is an element in such a finite field. The discrete logarithm in that finite field must be hard enough to prevent from attacks in a given time (e.g.$10\$ ...