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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 The method is to generalize unpublished calculations of Greenberg and Tilouine.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: 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]

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: 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: 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: 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: 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]

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]

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]