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Documents  11N36 | enregistrements trouvés : 19

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Research talks;Number Theory

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao. All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long ...

11N05 ; 11N35 ; 11N36

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- x; 496 p.
ISBN 978-2-85629-351-5

Astérisque , 0348

Localisation : Périodique 1er étage

Algèbres de von Neumann # analyse complexe discrète # asymétrie # asymptotique # champs # cohomologie complétée # compatibilité local-global # conjecture d'André-Oort # conjecture de Bogomolov # conjecture de Serre II # conjecture de Zilber-Pink # connexité rationnelle # corps convexes # courbes pseudo-holomorphes # creusage spectral # décomposition de John approchée # Déterminant jacobien # diffusion # espaces de Sobolev # équation de Schrödinger nonlinéaire # équations aux dérivées partielles # équirépartition # existence # fibration de Hitchin # fonction modulaire # fonctions BV # formule des traces # géométrie algébrique réelle # géométrie symplectique réelle # graphes expanseurs # groupes d'homotopie stable # Groupes non-moyennables # hauteurs # inégalité de Sobolev à trace # intersections exceptionnelles # invariance conforme # invariant de Kervaire # invariants de Gromov-Witten # inversibilité restreinte # lemme fondamental # mécanique des fluides # méthodes de crible # modèle d'Ising # multiplication complexe # obstruction de Brauer # percolation # physique statistique # points spéciaux # problème de Lehmer # problèmes énumératifs # propriét'e (tau) # réduction dimensionnelle # relations d'équivalence mesurées # spectres en anneaux structurés # surfaces K3 # théorème de Torelli global # théorie de l'homotopie chromatique # théorie de l'homotopie stable équivariante # théorie du contrôle # théorie symplectique des champs # théories o-minimales # transport de Brenier # variétés de Shimura # Variétés hyperkählériennes # variétés semi-abéliennes Algèbres de von Neumann # analyse complexe discrète # asymétrie # asymptotique # champs # cohomologie complétée # compatibilité local-global # conjecture d'André-Oort # conjecture de Bogomolov # conjecture de Serre II # conjecture de Zilber-Pink # connexité rationnelle # corps convexes # courbes pseudo-holomorphes # creusage spectral # décomposition de John approchée # Déterminant jacobien # diffusion # espaces de Sobolev # équation de ...

93C20 ; 35Q30 ; 11N05 ; 11N35 ; 11N36 ; 20F69 ; 05C25 ; 55Q45 ; 60K35 ; 82B20 ; 52C26 ; 81T40 ; 11S37 ; 11F70 ; 11F80 ; 22E55 ; 11G10 ; 11G50 ; 14K15 ; 65F50 ; 15A63 ; 46B07 ; 26A45 ; 53A10 ; 49Q15 ; 28A75 ; 14N10 ; 14N35 ; 14P99 ; 53D35 ; 53D45 ; 11G18 ; 03C64 ; 11E72 ; 14G05 ; 37A20 ; 20E05 ; 20P05 ; 46L10 ; 53C26 ; 14J28 ; 32J27 ; 46E35 ; 35Q55 ; 35B40 ; 35P25

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- 367 p.
ISBN

Astérisque , 0252

Localisation : Périodique 1er étage

Convergence des variétés # invariant passant à la limite # Courbure de Ricci et topologie # courbure presque positive # approximation de Haussdorff # volume # stabilité du volume # volume relativement petit # Epsilon-presque-partout # théorème de la sphère # théorème de comparaison # Bishop Gromov # Toponogov L2 # fonction distance # presque-harmonicité # presque parallélisme L2 # formule de Bochner # application d'Albanese # fonction zéta en caractéristique positive # module de Drinfeld # quasicristaux # pavage quasipériodique #règle d'incidence # W-algèbre # structure symplectique # équation intégrable # transformation de Darboux # opérateur vertex # distribution du spectre de matrice stochastique # K-théorie # conjecture de Novikov # conjecture de Baum-Connes # crible # crible asymptotique # grand crible # représentation des nombres premier par des polynnômes # entier Gaussien # équidistribution des racines des polynômes # quantification # variété de Poisson # star-produit # sous-variété symplectique # variété presque complexe # fibré vectoriel complexe # section hyperplane # pinceau de Lefschtz # lemme de Sard effective # base cenonique # totale positivité # groupe semisimple # cellule de Bruhat # polynôme de Kazhdan-Lusztig Convergence des variétés # invariant passant à la limite # Courbure de Ricci et topologie # courbure presque positive # approximation de Haussdorff # volume # stabilité du volume # volume relativement petit # Epsilon-presque-partout # théorème de la sphère # théorème de comparaison # Bishop Gromov # Toponogov L2 # fonction distance # presque-harmonicité # presque parallélisme L2 # formule de Bochner # application d'Albanese # fonction zéta en ...

53C23 ; 53C20 ; 53C21 ; 52A40 ; 52A38 ; 58E20 ; 58Cxx ; 58E35 ; 20C20 ; 17B37 ; 11G09 ; 11R58 ; 11T55 ; 82D25 ; 51M20 ; 52C17 ; 52C20 ; 52C22 ; 15A52 ; 35Q53 ; 47N30 ; 60H25 ; 22E70 ; 58G37 ; 47G30 ; 14H30 ; 35L20 ; 35B27 ; 78A05 ; 11L20 ; 11N32 ; 11N35 ; 11N36 ; 11N75 ; 81S10 ; 53C15 ; 58F05 ; 32L99 ; 32J25 ; 26C99 ; 14P10 ; 15A18 ; 14M15 ; 22E46 ; 14L30 ; 15A42 ; 58G32 ; 22E30 ; 22C05 ; 20G20 ; 14N10 ; 14H10 ; 14E99 ; 39A10 ; 11Fxx ; 11Gxx ; 13Nxx ; 12H05

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Research talks;Number Theory

This talk will survey some recent directions in the study of prime numbers that rely on bounds of exponential sums and advances in sieve theory. I will also describe some new results on the Riemann zeta function and Dirichlet functions, and pose some open problems.

11L20 ; 11N05 ; 11L07 ; 11N36 ; 11S40

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- xvii, 293 p.
ISBN 978-0-521-88851-6

Cambridge tracts in mathematics , 0175

Localisation : Collection 1er étage

grand crible # corps fini # groupe discret # somme exponentielle # fonction zéta # courbe algébrique # matrice unimodulaire # variété aléatoire

11-02 ; 11L07 ; 11N35 ; 11N36 ; 11T23 ; 14G10 ; 14G15 ; 11M41

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- 100 p.

Elemente der mathematik vom höherenstandpunkt aus , 0002

Localisation : Ouvrage RdC (TROS)

conjecture de Goldbach # critère général de primalité # fonction de la théorie des nombres # méthode du crible # nombre premier # nombre premier spécial # pi (x) et p indice n # preuve du théorème des nombres premiers # preuve du théorème sur la progression arithmétique # somme de premiers

11A41 ; 11A51 ; 11Axx ; 11N35 ; 11N36

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- 362 p.
ISBN 978-0-691-12437-7

London mathematical society monographs series

Localisation : Ouvrage RdC (HARM)

théorie des nombres # nombres premiers # cribles # méthode du crible # applications

11-02 ; 11A41 ; 11N35 ; 11N36

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- xx; 527 p.
ISBN 978-0-8218-4970-5

American mathematical society colloquium publications , 0057

Localisation : Collection 1er étage

théorie des nombres # cribles

11N35 ; 11N36 ; 11N05 ; 11N13 ; 11N32 ; 11N37 ; 11J71 ; 11E25

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- x; 201 p.
ISBN 978-81-85931-90-6

Harish-Chandra research institute lecture notes , 0001

Localisation : Ouvrage RdC (RAMA)

nombres premiers # séries # cribles # crible de Selberg # arithmétique

11N35 ; 11-02 ; 11N36

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- 122 p.
ISBN 978-0-521-20915-1

Cambridge tracts in mathematics , 0070

Localisation : Collection 1er étage

crible # estimation # filtration # filtration asymtotique # nombres premiers # théorie des nombres

11A41 ; 11N35 ; 11N36 ; 11N37

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- viii; 297 p.
ISBN 978-0-8218-4406-9

CRM proceedings & lecture notes , 0046

Localisation : Collection 1er étage

théorie des nombres # nombre entier # nombre lisse # fonction arithmétique # théorème de Erdös-Kac # polynôme cyclotomique # forme quadratique # fonction zêta # série de Diriclet # L-fonction

11-06 ; 11N25 ; 11N36 ; 11N60 ; 11L07 ; 11L20 ; 11L40 ; 11N37 ; 11N56 ; 11P83

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- 224 p.
ISBN 978-0-521-84816-9

London mathematical society student texts , 0066

Localisation : Collection 1er étage

courbe elliptique # théorème de densité # méthode de crible # application des méthodes de crible # théorème de Bombieri-Vinogradov

11G05 ; 11N36 ; 11R45

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- xxi; 266p.
ISBN 978-0-521-89487-6

Cambridge tracts in mathematics , 0177

Localisation : Collection 1er étage

crible # méthode du crible # application

11N35 ; 11N36

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Research talks;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...

11G05 ; 11N36 ; 11F80

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Research talks;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...

11G05 ; 11N36 ; 11F80

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Research talks;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...

11G05 ; 11N36 ; 11F80

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- 171 p.

Localisation : Ouvrage RdC (DART)

distribution de nombres entiers # méthode de crible # valeur de polynômes # crible pondéré # polynôme

11N25 ; 11N36 ; 11N32

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- 101 p.

Localisation : Ouvrage RdC (RIVA)

nombres premiers # sommes d'exponentielles # estimation de sommes d'exponentielles # répartition des nombres premiers # cribles # applications des méthodes de cribles

11A41 ; 11L03 ; 11L07 ; 11N05 ; 11N35 ; 11N36

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- 133 p.

Localisation : Ouvrage RdC (SAIA)

crible # équation différentielle aux différences finies # méthode du col # théorie analytique des nombres # transformation de Laplace

11N36 ; 44A10 ; 11Mxx

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