Contre-exemples au principe de Hasse pour les courbes de Fermat

Alain Kraus

Displaying similar documents to “Contre-exemples au principe de Hasse pour les courbes de Fermat”

Une note à propos du jacobien de n fonctions holomorphes à l'origine de ℂⁿ

M. Hickel (2008)

Annales Polonici Mathematici

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Let f₁,...,fₙ be n germs of holomorphic functions at the origin of ℂⁿ, such that $f_{i} (0) = 0$ , 1 ≤ i ≤ n. We give a proof based on J. Lipman’s theory of residues via Hochschild homology that the jacobian of f₁,...,fₙ belongs to the ideal generated by f₁,...,fₙ if and only if the dimension of the germ of common zeros of f₁,...,fₙ is strictly positive. In fact, we prove much more general results which are relative versions of this result replacing the field ℂ by convenient noetherian rings A (Ths....

Équations elliptiques non linéaires monotones avec un deuxième membre $L^{1}$ ou mesure

François Murat (1998)

Journées équations aux dérivées partielles

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On considère le problème : $\{\begin{matrix} - div a (x, D u) = f dans Ω, \\ u = 0 sur \partial Ω, \end{matrix}$ où $Ω$ est un ouvert borné de $𝐑^{N}$ , où $a (x, ξ)$ est une fonction de Carathéodory, monotone en $ξ$ , coercive, qui définit un opérateur dans $W_{0}^{1, p} (Ω)$ (avec $1 < p \leq N$ ), et où $f$ appartient à $L^{1} (Ω)$ ou est une mesure bornée sur $Ω$ . On introduit une nouvelle définition de la solution de ce problème, la notion de solution renormalisée (ou entropique), et on montre l’existence d’une telle solution et sa continuité par rapport à $f$ . Quand $f$ appartient à $L^{1} (Ω)$ , on montre en outre...

Généralisation d'un théorème de Haagerup

Ferdaous Kellil, Guy Rousseau (2005)

Studia Mathematica

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Let G be a group of automorphisms of a tree X (with set of vertices S) and H a kernel on S × S invariant under the action of G. We want to give an estimate of the $l^{r}$ -operator norm (1 ≤ r ≤ 2) of the operator associated to H in terms of a norm for H. This was obtained by U. Haagerup when G is the free group acting simply transitively on a homogeneous tree. Our result is valid when X is a locally finite tree and one of the orbits of G is the set of vertices at even distance from a given...

Propagation de la 2-birationalité

Claire Bourbon, Jean-François Jaulent (2013)

Acta Arithmetica

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Let L/K be a 2-birational CM-extension of a totally real 2-rational number field. We characterize in terms of tame ramification totally real 2-extensions K’/K such that the compositum L’=LK’ is still 2-birational. In case the 2-extension K’/K is linearly disjoint from the cyclotomic ℤ₂-extension $K^{c} / K$ , we prove that K’/K is at most quadratic. Furthermore, we construct infinite towers of such 2-extensions.

Complexité et automates cellulaires linéaires

Valérie Berthé (2010)

RAIRO - Theoretical Informatics and Applications

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The aim of this paper is to evaluate the growth order of the complexity function (in rectangles) for two-dimensional sequences generated by a linear cellular automaton with coefficients in $ℤ / l ℤ$ , and polynomial initial condition. We prove that the complexity function is quadratic when is a prime and that it increases with respect to the number of distinct prime factors of .

Courbes elliptiques sur ℚ, ayant un point d’ordre 2 rationnel sur ℚ, de conducteur $2^{N} p$

Wilfrid Ivorra

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Let p be a prime number ≥ 29 and N be a positive integer. In this paper, we are interested in the problem of the determination, up to ℚ-isomorphism, of all the elliptic curves over ℚ whose conductor is $2^{N} p$ , with at least one rational point of order 2 over ℚ. This problem was studied in 1974 by B. Setzer in case N = 0. Consequently, in this work we are concerned with the case N ≥ 1. The results presented here are analogous to those obtained by B. Setzer and allow one in practice to find...

Plus grand facteur premier de valeurs de polynômes aux entiers

R. de la Bretèche (2015)

Acta Arithmetica

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Let P⁺(n) denote the largest prime factor of the integer n. Using the Heath-Brown and Dartyge methods, we prove that for any even unitary irreducible quartic polynomial Φ with integral coefficients and the associated Galois group isomorphic to V₄, there exists a positive constant $c_{Φ}$ such that the set of integers n ≤ X satisfying $P ⁺ (Φ (n)) \geq X^{1 + c_{Φ}}$ has a positive density. Such a result was recently proved by Dartyge for Φ(n) = n⁴ - n² + 1. There is an appendix written with Jean-François Mestre. ...

Quantification pour les paires symétriques et diagrammes de Kontsevich

Alberto S. Cattaneo, Charles Torossian (2008)

Annales scientifiques de l'École Normale Supérieure

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In this article we use the expansion for biquantization described in [7] for the case of symmetric spaces. We introduce a function of two variables $E (X, Y)$ for any symmetric pairs. This function has an expansion in terms of Kontsevich’s diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this $E$ function. We prove that Cattaneo and Felder’s star product coincides with Rouvière’s for any symmetric pairs. We generalize some of...

Sur une application de la formule de Selberg-Delange

F. Ben Saïd, J.-L. Nicolas (2003)

Colloquium Mathematicae

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E. Landau has given an asymptotic estimate for the number of integers up to x whose prime factors all belong to some arithmetic progressions. In this paper, by using the Selberg-Delange formula, we evaluate the number of elements of somewhat more complicated sets. For instance, if ω(m) (resp. Ω(m)) denotes the number of prime factors of m without multiplicity (resp. with multiplicity), we give an asymptotic estimate as x → ∞ of the number of integers m satisfying $2^{ω (m)} m \leq x$ , all prime factors...

Théorèmes de préparation Gevrey et étude de certaines applications formelles

Augustin Mouze (2003)

Annales Polonici Mathematici

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We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps F from $ℂ^{s}$ to $ℂ^{p}$ without constant term such that the rank of the Jacobian matrix of F is equal to 1. Let be a formal power series. If F is a holomorphic map, the following result is well known: ∘ F is analytic implies there exists a convergent...

Espaces de suites réelles complètement métrisables

Pierre Casevitz (2001)

Fundamenta Mathematicae

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Let X be an hereditary subspace of the Polish space $ℝ^{ω}$ of real sequences, i.e. a subspace such that [x = (xₙ)ₙ ∈ X and ∀n, |yₙ| ≤ |xₙ|] ⇒ y = (yₙ)ₙ ∈ X. Does X admit a complete metric compatible with its vector structure? We have two results: ∙ If such an X has a complete metric δ, there exists a unique pair (E,F) of hereditary subspaces with E ⊆ X ⊆ F, (E,δ) complete separable, and F complete maximal in a strong sense. On E and F, the metrics have a simple form, and the spaces E are...

Les types de données syntaxiques du système $ℱ$

Samir Farkh, Karim Nour (2010)

RAIRO - Theoretical Informatics and Applications

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We give in this paper a purely syntactical definition of input and output types of system $ℱ$ . We define the syntactical data types as input and output types. We show that any type with positive quantifiers is a syntactical data type and that an input type is an output type. We give some restrictions on the -elimination rule in order to prove that an output type is an input type.

La structure des sous-espaces de treillis

José L. Marcolino Nhani

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We study some geometrical properties of a new structure introduced by G. Pisier: the structure of lattice subspaces. We show first that if X and Y are Banach lattices such that $B_{r} (X, Y) = B (X, Y)$ , then X is an AL-space or Y is an AM-space. We introduce the notion of homogeneous lattice subspace and we show that up to regular isomorphism, the only homogeneous lattice subspace of $L^{p} (Ω, μ)$ , for 2≤ p < ∞, is G(I). We also show a version of the Dvoretzky theorem for this structure. We end this paper by giving...

Ground states of supersymmetric matrix models

Gian Michele Graf (1998-1999)

Séminaire Équations aux dérivées partielles

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We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the $d = 9$ model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in $d = 9$ . Moreover, it would be unique. Other values of $d$ , where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation....

Fonctions digitales le long des nombres premiers

Bruno Martin, Christian Mauduit, Joël Rivat (2015)

Acta Arithmetica

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In a recent work we gave some estimations for exponential sums of the form $\sum_{n \leq x} Λ (n) e x p (2 i π (f (n) + β n))$ , where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.

Derivees tangentielles des fonctions de la classe $^{k, α}$ dans les domaines de type fini de ℂ²

Laurent Verdoucq (2002)

Annales Polonici Mathematici

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Let Ω be a domain of finite type in ℂ² and let f be a function holomorphic in Ω and belonging to $^{k, α} (Ω ̅)$ . We prove the existence of boundary values for some suitable derivatives of f of order greater than k. The gain of derivatives holds in the complex-tangential direction and it is precisely related to the geometry of ∂Ω. Then we prove a property of non-isotropic Hölder regularity for these boundary values. This generalizes some results given by J. Bruna and J. M. Ortega for the unit ball. ...

Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité

Colette Moeglin (2002)

Journal of the European Mathematical Society

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In this paper, we are giving parameters for discrete series of classical $p$ -adic groups. We first define: the analogous of the Langlands morphism of $W_{F}$ in the $L$ -group, part of the analogous of the character of the centralizer of that morphism and, to supply the missing part of the full definition of that character, the cuspidal support of the representation. Then, we state an hypothesis on the reducibility points for induced of cuspidal representations. And we prove that, under this hypothesis,...

Théorème des nombres premiers pour les fonctions digitales

Bruno Martin, Christian Mauduit, Joël Rivat (2014)

Acta Arithmetica

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The aim of this work is to estimate exponential sums of the form $\sum_{n \leq x} Λ (n) e x p (2 i π (f (n) + β n))$ , where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.