Sinnou David - Academia.edu (original) (raw)

Papers by Sinnou David

Acta Arithmetica, 2022

For a given rational number x and an integer s ≥ 1, let us consider a generalized polylogarithmic... more For a given rational number x and an integer s ≥ 1, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by Φs(x, z) = ∞ k=0 z k+1 (k + x + 1) s. We prove the linear independence over any number field K of the numbers 1 and Φs j (xj, αi) with any choice of distinct shifts x1,. .. , x d with 0 ≤ x1 <. .. < x d < 1, as well as any choice of depths 1 ≤ s1 ≤ r1,. .. , 1 ≤ s d ≤ r d , at distinct algebraic numbers α1,. .. , αm ∈ K subject to a metric condition. As is usual in the theory, the points αi need to be chosen sufficiently close to zero with respect to a given fixed place v0 of K, Archimedean or finite. This is the first linear independence result with distinct shifts x1,. .. , x d that allows values at different points for generalized polylogarithmic functions. Previous criteria were only for the functions with one fixed shift or at one point. Further, we establish another linear independence criterion for values of the generalized polylogarithmic function with cyclic coefficients. Let q ≥ 1 be an integer and a = (a1,. .. , aq) ∈ K q be a q-tuple whose coordinates supposed to be cyclic with the period q. Consider the generalized polylogarithmc function with coefficients Φa,s(x, z) = ∞ k=0 a k+1 mod (q) • z k+1 (k + x + 1) s. Under suitable condition, we show that the values of these functions are linearly independent over K. Our key tool is a new non-vanishing property for a generalized Wronskian of Hermite type associated to our explicit constructions of Padé approximants for this family of generalized polylogarithmic function.

HAL (Le Centre pour la Communication Scientifique Directe), 2014

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us a... more Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L * (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.

HAL (Le Centre pour la Communication Scientifique Directe), 2021

arXiv (Cornell University), Oct 19, 2020

Let r, m be positive integers. Letx be a rational number with 0 ≤ x < 1. Consider Φs(x, z) = ∞ k=... more Let r, m be positive integers. Letx be a rational number with 0 ≤ x < 1. Consider Φs(x, z) = ∞ k=0 z k+1 (k + x + 1) s the s-th Lerch function with s = 1, 2, • • • , r. When x = 0, this is a polylogarithmic function. Let α1, • • • , αm be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with 0 < |αj | < 1 (1 ≤ j ≤ m). In this article, we show a criterion for the linear independence, over an algebraic number field containing Q(α1, • • • , αm), of all the rm + 1 numbers : Φ1(x, α1), Φ2(x, α1), • • • , , Φr(x, α1), Φ1(x, α2), Φ2(x, α2), • • • , Φr(x, α2), • • • • • • , Φ1(x, αm), Φ2(x, αm), • • • , Φr(x, αm) and 1. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [9].

Annales mathématiques du Québec, 2019

nous étudions des conjectures de Bertrand et Rodriguez-Villegas sur le comportement du covolume d... more nous étudions des conjectures de Bertrand et Rodriguez-Villegas sur le comportement du covolume de sous réseaux du groupe des unités (via le plongement logarithmique) et montrons que ces conjectures sont vraies dans plusieurs cas. Ces résultats prolongent nos résultats antérieurs ainsi que ceux de nombreux auteurs dont récemment Chinburg-Friedman-Sundstrom. Abstract : we discuss two conjectures of Bertrand and Rodriguez-Villegas on the behavior of the (co)volume of subgroups of units (via the logarithmic embedding). We show several results in the direction of these conjectures. They extend some previous results of ourselves and others, among them recently Chinburg-Friedman-Sundstrom.

C R Acad Sci Ser I Math, 1998

Int J Number Theory, 2008

International Mathematics Research Papers, 2007

The Ramanujan Journal, 2001

C R Acad Sci Ser I Math, 2000

Http Www Theses Fr, 1989

Dans une premiere partie, nous raffinons un resultat de d. Masser sur l'orbite galoisienne d&... more Dans une premiere partie, nous raffinons un resultat de d. Masser sur l'orbite galoisienne d'un point de torsion d'une variete abelienne simple, principalement polarisee, definie sur un corps de nombres, en precisant la dependance en la variete de la minoration. Ce resultat est obtenu a l'aide d'estimations precises de la croissance analytique des fonctions theta ainsi que sur les proprietes de leurs derivees algebriques (construites par c. Shimura). Dans une deuxieme partie, nous precisons un resultat de p. Philippon et m. Waldschmidt sur les minorations de formes lineaires de logarithmes de points algebriques d'un groupe algebrique commutatif. Outre une extension a une representation de l'exponentielle du groupe des resultats de la premiere partie, (valables pour les fonctions theta de base), il s'agit de preciser la dependance en le groupe des formules de translations. Un critere de generation normale est utilise en plus des techniques modulaires developpees dans la premiere partie

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Journal of the Institute of Mathematics of Jussieu

International Journal of Number Theory

According to F. Amoroso and R. Dvornicich [J. Number Theory 80, No. 2, 260–272 (2000; Zbl 0973.11... more According to F. Amoroso and R. Dvornicich [J. Number Theory 80, No. 2, 260–272 (2000; Zbl 0973.11092)], an algebraic number in the maximal abelian extension of ℚ which is not a root of unity has an absolute logarithmic height at least (log5)/12. The main theorem of this paper is an analog of this result in the context of Drinfeld modules of rank ≥1. The canonical height on such a Drinfeld module has been introduced by L. Denis [Math. Ann. 294, No. 2, 213–223 (1992; Zbl 0764.11027)]. The authors show the existence of a lower bound for the canonical height of any non-torsion point on an abelian extension of the ground field. Among the tools which are developed in this paper is a lifting of the Frobenius morphism for modules having complex multiplication, which seems to be new for Drinfeld modules of arbitrary rank.