Des résultats d'irrationalité pour deux fonctions particulières.
We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.
Soit un nombre premier impair. Soit une -extension galoisienne de corps ne contenant pas les racines -ièmes de l’unité : . Notons le groupe de Galois de et son sous-groupe de Frattini. Via une notion de descente galoisienne et les parallélogrammes galoisiens qu’elle induit, nous construisons ici toutes les extensions telles que soit d’ordre .
Un article précédent paru dans le Séminaire de Théorie des Nombres de Bordeaux contient une description détaillée des orbites de voisines pour les représentants des 15 classes de formes parfaites à 7 variables, non équivalentes à et qui possèdent plus de 28 vecteurs minimaux. Le lecteur trouvera ici le résultat correspondant pour , ainsi qu’une description plus détaillée des voisines de . Ceci termine la classification des formes parfaites en dimension 7. Un premier pas en direction de la classification...
The notion of designs in Grassmannian spaces was introduced by the author and R. Coulangeon, G. Nebe, in [3]. After having recalled some basic properties of these objects and the connections with the theory of lattices, we prove that the sequence of Barnes-Wall lattices hold -Grassmannian designs. We also discuss the connections between the notion of Grassmannian design and the notion of design associated with the symmetric space of the totally isotropic subspaces in a binary quadratic space, which...
This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this...
Dati o numeri algebrici non nulli tali che non è una radice dell'unità per ogni , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro , al variare di in , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni in posizione generica dell'equazione polinomiale-esponenziale disomogenea non supera una costante esplicita dipendente solo da .
Let ℱn = circ (︀F*1 , F*2, . . . , F*n︀ be the n×n circulant matrix associated with complex Fibonacci numbers F*1, F*2, . . . , F*n. In the present paper we calculate the determinant of ℱn in terms of complex Fibonacci numbers. Furthermore, we show that ℱn is invertible and obtain the entries of the inverse of ℱn in terms of complex Fibonacci numbers.
Let be a finite subset of a partially ordered set . Let be an incidence function of . Let denote the matrix having evaluated at the meet of and as its -entry and denote the matrix having evaluated at the join of and as its -entry. The set is said to be meet-closed if for all . In this paper we get explicit combinatorial formulas for the determinants of matrices and on any meet-closed set . We also obtain necessary and sufficient conditions for the matrices...
The aim of this paper is to study determinants of matrices related to the Pascal triangle.