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Using a new approach starting with a residue computation, we sharpen some of the known estimates for the counting function of friable integers. The improved accuracy turns out to be crucial for various applications, some of which concern fundamental questions in probabilistic number theory. [ABSTRACT FROM AUTHOR]

The article is devoted to Edouard Lucas's contribution to the development of mathematical recreations in the France of the post 1870 war period. Lucas's name is associated to four volumes of Récréations mathématiques published between 1882 and 1894 (the last two having been published posthumously) and to a posthumous volume L'Arithmétique amusante , which appeared in 1895. The author analyzes the context of reform of science education in relation to which mathematical recreations appeared as a means of attracting a wider public to scientific activities and inspiring young people to study science. The article brings to light how the milieu of new associations which took shape to promote science (Association Française pour l'Avancement des Sciences, Société Mathématique de France) allowed the constitution of social groups internationally connected and quite active in the promotion and development of mathematical recreations. Lastly, the article suggests that this type of mathematical activity allowed the cultivation of fields that at the time the French academic milieu perceived as marginal such as number theory and analysis situs as well as their applications. [ABSTRACT FROM AUTHOR]

Résumé: Nous étudions les propriétés locales des espaces de Berkovich sur  Z. À l’aide de théorèmes de Weierstraß, nous montrons que les anneaux locaux de ces espaces sont noethériens, réguliers dans le cas des espaces affines et excellents. Nous prouvons également que le faisceau structural est cohérent. Nos méthodes s’adaptent à d’autres anneaux de base (corps valués, anneaux de valuation discrète, anneaux d’entiers de corps de nombres, etc.) et traitent de façon unifiée espaces complexes et p-adiques. [ABSTRACT FROM AUTHOR]

Résumé: Dans [10], on a déterminé le spectre du Laplacien singulier attaché aux métriques canoniques sur . Le but de cet article est lʼétude de , la fonction Zêta associée à ce spectre. On montre quʼelle est prolongeable en une fonction holomorphe en zéro et on détermine les valeurs de et pour tout : [ABSTRACT FROM AUTHOR]

Abstract: We generalize to an arbitrary number field the result of Davenport and Schmidt about approximation to a real number and his square by rational numbers with the same denominator, as well as its p-adic analog due to Teulié. By an extension of Royʼs method, we prove that our new result is optimal. [Copyright &y& Elsevier]

Augustin-Louis Cauchy published most of his arithmetical research between 1829 and 1840. These are however only mentioned in some number theory history centered on reciprocity laws or on theory of algebraic numbers. They are described as containing some results similar to those of Gauss, Jacobi and Dirichlet but in a incomplete and disorganized way. The objective of this paper is to present an analysis of Cauchy's arithmetical texts published between 1829 and 1840 to show that they contain a rather consistent set of results related to quadratic forms 4pu = x2 + ny2, where p is a prime and n a divisor of p-1. We will also discuss the particular form of this body of texts and the strategy we used to find the guidelines of the work of Cauchy.

Abstract: We give an upper bound of the number of negative eigenvalues corresponding to an elliptic operator perturbed by a potential, and where the boundary conditions are Robin. [Copyright &y& Elsevier]

Abstract: We prove a formula for the derivative of the p-adic L-function associated with the symmetric square representation of an elliptic curve over a totally real field in which p is inert, under certain assumptions on the conductor. In particular, this proves a conjecture of Greenberg on trivial zeros. The method is to generalize unpublished calculations of Greenberg and Tilouine. [Copyright &y& Elsevier]

Abstract: Let k be a number field and its class group. Let Γ be a finite group and its order. Let (resp. ) be the subset of consisting of those classes which are realizable as Steinitz classes of Galois extensions (resp. tamely ramified Galois extensions) of k with Galois group isomorphic to Γ. In the present article, we suppose that Γ is realizable as Galois group over k of a Galois extension (resp. tame Galois extension) – e.g. Γ solvable – and the center of Γ is non-trivial – e.g. Γ nilpotent non-trivial. For each prime divisor p of the order of , we define a natural number . We show that if the class number of k is prime to , then (resp. ) is the full group . For instance, this result applies to a nilpotent group Γ having even order, with . [Copyright &y& Elsevier]

Abstract: We give an effective lower bound for the number of non-zero digits among the first N digits of the expansion of an irrational algebraic number in an integer base. [Copyright &y& Elsevier]

Résumé: Pour des niveaux particuliers (puissance croissante dʼun nombe premier fixé), nous avons déjà montré dans Rouymi (2011) une première minoration non triviale de la proportion de non-annulation des formes primitives au point critique. Il est en fait possible dʼaméliorer cette minoration et obtenir une proportion positive de non-annulation : cʼest lʼobjet de cet article. Notre résultat principal est une généralisation dʼun résultat obtenu obtenu par Kowalski, Michel et Vanderkam. [ABSTRACT FROM AUTHOR]

Abstract: This paper is about the pole of some Eisenstein series for classical groups over a number field. In a previous paper, we have shown how to normalize intertwining operators in such a way that they are holomorphic for positive parameters. Here we show that the image of such operators is (in the interesting cases) either 0 or an irreducible representation. This enables us to compute explicitly the residue of the Eisenstein series obtained from square integrable cohomological representations. At the end of the paper we give necessary and sufficient conditions in terms of Arthurʼs data in order that a square integrable cohomological representation is cuspidal; the conditions are not totally satisfactory and we explain what we expect when Arthurʼs results will be fully available. [Copyright &y& Elsevier]

Abstract: We give lower bounds for the number of effective divisors of degree of an algebraic function field of one variable of genus defined over a finite field. We deduce lower bounds for the class number, depending mainly on the number of places of a certain degree, which improve the best known lower bounds in many cases. Moreover, we prove that any family of algebraic function fields having asymptotically (with respect to the genus) a large number of places of a certain degree , have a large asymptotical class number for which we give an estimate. [Copyright &y& Elsevier]

Abstract: This Note deals with real continuous-time processes which admit a -valued autoregressive representation of order one (ARD(1)). Under some regularity conditions, we establish laws of large numbers, the central limit theorem and the law of the iterated logarithm for ARD(1). [Copyright &y& Elsevier]

Abstract: In this article we give an infinite number of good initial values for the Lucas–Lehmer sequence. [Copyright &y& Elsevier]

Abstract: Let k be a number field and its ring of integers. Let l be a prime number and m a natural number. Let C (resp. H) be a cyclic group of order l (resp. m). Let be a metacyclic group of order lm, with H acting faithfully on C. Let be a maximal -order in the semi-simple algebra containing , and its locally free class group. We define the set of realizable classes to be the set of classes such that there exists a Galois extension which is tame, with Galois group isomorphic to Γ, and for which , where is the ring of integers of N. In the present article, we define a subset of and prove, by means of a description using a Stickelberger ideal, that it is a subgroup of , under the hypothesis that k and the l-th cyclotomic field over are linearly disjoint. [Copyright &y& Elsevier]

Résumé: Dans ce papier, on rectifie sous certaines hypothèses le théorème principal (Théorème 2, page 52) de l''article ci-dessus en donnant une formule arithmétique explicite des coefficients de Li, . [Copyright &y& Elsevier]

Abstract: As a corollary of a more general result, we present new rational approximations to the value of the Gamma function at a given rational point, where and are solutions of a linear recurrence of order 3 with polynomials coefficients. [Copyright &y& Elsevier]

Résumé: Nous exprimons les valeurs spéciales de fonctions L en caractéristique p en termes de groupes d''extension dans la catégorie des chtoucas, en utilisant un peu de théorie de Fontaine en caractéristique p. [Copyright &y& Elsevier]

Abstract: This article describes a proof based on the properties of p-adic periods of Fontaine''s theorem that states that a -linear, continuous, Galois-equivariant map from into itself is a homothecy. [Copyright &y& Elsevier]

Abstract: The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework. In the case of a Galois extension of a number field with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal class group of F by the Tate cohomology group . The Mayer–Vietoris exact sequence enables us to describe in a quite explicit way a quotient of the subgroup of the ideal class group , where N is the norm. We also prove a short exact sequence where is the group of semi-local units of F. Finally, we conclude this paper by applications of our methods to Number Theory. [Copyright &y& Elsevier]

Abstract: Let θ be a Pisot number less than 2, m a positive rational integer, and the set of the polynomials with coefficients in evaluated at θ. We give a lower bound for the greatest limit point of common differences of consecutive elements of . To cite this article: T. Zaïmi, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and , where m is a positive integer. Let be a qth primitive root of 1 and be the ring of integers in . In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372–384] I. Gaál and L. Robertson show that if , where is the class number of , then if is a generator of (in other words ) either α is equals to a conjugate of an integer translate of or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if is a generator of then either α is a conjugate of an integer translate of or is an odd integer. [Copyright &y& Elsevier]

Abstract: In this paper, we are interested in diophantine equations of type where F is a separable homogeneous form of degree ⩾3 with integer coefficients, d a fixed integer ⩾1 and p a prime number ⩾7. As a consequence of the abc conjecture, if p is sufficiently large and is a nontrivial proper solution of the above equation, we have . In the case where F has degree 3, we associate to an elliptic curve defined over called the Frey curve or Hellegouarch–Frey curve. This allows us to deduce our conjecture from another one about elliptic curves attributed to G. Frey and B. Mazur (which is itself a consequence of the abc conjecture). We then applied our construction to the study of an explicit form. We give some results about the set of nontrivial proper solutions of the equation considered for several values of d. [Copyright &y& Elsevier]

Abstract: We prove, using the main result of a previous Note by the author, that the Eisenstein classes (defined there in Section 4) of Hilbert–Blumenthal families degenerate, at the ∞ cusp of the Baily–Borel compactification of the base, to a special value of an L-function of the underlying totally real number field. As a corollary we get both an alternative proof of the Klingen–Siegel theorem and a non-vanishing result for some of these classes. To cite this article: D. Blottière, C. R. Acad. Sci. Paris, Ser. I 345 (2007). [Copyright &y& Elsevier]

Abstract: Imitating the Lubin''s philosophy on nonarchimedean dynamical systems, we prove that every finite family of inversibles series of which commute for the law ○ is connected with a finite family of automorphisms of a formal group over . In some cases these formal groups are reduction on of Lubin–Tate formal groups over finite extensions of . [Copyright &y& Elsevier]

Abstract: Using geometric tools introduced by P. Cohen, H. Shiga, J. Wolfart and G. Wüstholz, we show in Theorem 1 that when a certain Gauss hypergeometric function takes an algebraic value at an algebraic point, then another Gauss hypergeometric function takes a transcendental value at a related algebraic point. Using Appell hypergeometric functions, which generalize to two variables the Gauss functions, we study values at algebraic points of a new transcendental function defined in terms of these two functions. By Theorem 2, these values correspond to abelian varieties in the same isogeny class. Using a result of Edixhoven–Yafaev [B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. of Math. 157 (2003) 621–645], this last result is in turn related to the distribution of the moduli of such abelian varieties in certain Shimura varieties. [Copyright &y& Elsevier]

Abstract: We generalise Li''s criterion, already known for the Riemann zeta function, to a large class of Dirichlet series. We give first an explicit formula for the coefficients , for all positive integers n and ρ runs over all the non-trivial zeros of a function F in this class. To do so, we use the Weil Explicit Formula. [Copyright &y& Elsevier]

Abstract: We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this -module to the homology of the modular curve . We give an interpretation of Gross formula for special values of L-functions of modular forms in this context. As an application, we obtain a formula for the sum of the squares of the optimal embeddings numbers of quadratic orders in a definite quaternion algebra. [Copyright &y& Elsevier]

Abstract: We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder- function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: We study the ℓ-part of the wild étale kernels of an arbitary number field F for a given prime ℓ in connection with the logarithmic ℓ-class groups . From the logarithmic arithmetic we deduce rank formulas, periodicity and reflection theorems, triviality characterizations and various consequences. [Copyright &y& Elsevier]

Abstract: In this Note, we construct for some normal surface X an object of whose homology sheaves are not strictly -invariant. This disproves the -connectivity conjecture of F. Morel. To cite this article: J. Ayoub, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Using an inequality for the Weil height on an Abelian extension of the rationals and a theorem of Linnik, we prove a lower bound for the index of the annihilator of the ideal class group of a cyclotomic field. This lower bound is exponential in the degree of the field. To cite this article: F. Amoroso, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: We present some results on projective toric varieties which are relevant in diophantine geometry. We interpret and study several invariants attached to these varieties by geometrical and combinatorial terms. We also give a Bézout theorem for Chow weights of projective varieties and an application to the theorem of successive algebraic minima. These results are extracted from the two texts of Philippon and Sombra mentioned in the references at the end of this Note (both downloadable from http://fr.arxiv.org). To cite this article: P. Philippon, M. Sombra, C. R. Acad. Sci. Paris, Ser. I 340 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we prove, in the p-adic case, a 0–1 law about the linearization of families under a polynomial perturbation in t. This gives new examples of non linearizable functions in the p-adic case. To cite this article: D. Vieugué, C. R. Acad. Sci. Paris, Ser. I 339 (2004). [Copyright &y& Elsevier]

In this article we study a generalization of the notion of Pfister neighbors. An anisotropic quadratic form φ over a field F of characteristic not 2 is called a quasi-Pfister neighbor when the anisotropic part (φF(φ))an is F(φ)-similar to an F-quadratic form φ where F(φ) denotes the function field of the projective quadric given by φ. We prove the uniqueness of φ up to F-similarity for forms φ of dimension ≤ 8, odd dimension and many others of large dimension, and in these cases we give a precise description of ψ. [ABSTRACT FROM AUTHOR]

Let (X,ω) be a compact complex Hermitian manifold, and let T⩾γ be a d-closed (1,1) almost positive current on X. A variant of Demailly''s regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents Tm with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C∞, the regularizing currents Tm can be chosen such that Tm⩾γ−C/m for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

We give an explicit example of an Enriques surface without a rational point over C((t)). To cite this article: G. Lafon, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

In this Note, we study the behavior of the Hardy–Littlewood maximal function M on cusp manifolds in terms of the growth of the volume of the base space. In particular, we prove that for all 1 fixed, there exists such a manifold on which M is bounded on Lp for p>p0 but not for 1⩽p. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

Let N be the set of positive integers and A a subset of N. For n∈N, let p(A,n) denote the number of partitions of n with parts in A. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N∈N and B⊂{1,2,…,N}, there is a unique set A=A0(B,N), such that p(A,n) is even for n>N. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered σ(A,n)=∑d∣n,d∈Ad, and proved that for all k≥0, the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f(0)=1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in F2[z] with P(0)=Q(0)=1, then for all k≥0 the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. [Copyright &y& Elsevier]

We show that any binary type II code can be built as a Cortex code where the basis code is the [8,4,4] extended Hamming code. To cite this article: A. Otmani, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

After introducing the notion of group automorphism hyperbolic relative to a family of subgroups, we establish an analog of the Bestvina–Feighn''s Combination Theorem for mapping-tori groups Gα=G⋊αZ of relatively hyperbolic automorphisms α of hyperbolic groups G. Both Farb''s and Gromov''s relative hyperbolicity are considered. To cite this article: F. Gautero, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

Abstract: We give an elementary counter-example for a conjecture of Pineiro, Szpiro and Tucker about the uniform distribution of algebraic numbers with small height. To cite this article: P. Autissier, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]