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Descent via (3,3)-isogeny on Jacobians of genus 2 curves

Nils Bruin, E. Victor Flynn, Damiano Testa (2014)

Acta Arithmetica

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.

Descente et parallélogramme galoisiens

Richard Massy, Sylvie Monier-Derviaux (1999)

Journal de théorie des nombres de Bordeaux

Soit p un nombre premier impair. Soit D / J une p -extension galoisienne de corps ne contenant pas les racines p -ièmes de l’unité : J ∩ μ p = 1 . Notons G le groupe de Galois de D / J et Φ ( G ) son sous-groupe de Frattini. Via une notion de descente galoisienne et les parallélogrammes galoisiens qu’elle induit, nous construisons ici toutes les extensions D / J telles que Φ ( G ) soit d’ordre p .

Description des voisines de E 7 , D 7 , D 8 , et D 9

David-Olivier Jaquet-Chiffelle (1992)

Journal de théorie des nombres de Bordeaux

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 E 7 et qui possèdent plus de 28 vecteurs minimaux. Le lecteur trouvera ici le résultat correspondant pour E 7 , ainsi qu’une description plus détaillée des voisines de D 7 . Ceci termine la classification des formes parfaites en dimension 7. Un premier pas en direction de la classification...

Designs, groups and lattices

Christine Bachoc (2005)

Journal de Théorie des Nombres de Bordeaux

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 6 -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...

Determinant Representations of Sequences: A Survey

A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy (2014)

Special Matrices

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...

Determinanti polinomiali-esponenziali

Raffaele Marcovecchio (2004)

Bollettino dell'Unione Matematica Italiana

Dati m = 2 o m = 3 numeri algebrici non nulli α = α 1 , … , α m tali che α j / α l non è una radice dell'unità per ogni j ≠ l , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro G ⁢ a ; x , al variare di x in Z 4 , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni y ∈ Z 3 in posizione generica dell'equazione polinomiale-esponenziale disomogenea G ⁢ a ; 0 , y = 0 non supera una costante esplicita N ⁢ d dipendente solo da d = [ Q ( α 1 , … , α m ) : Q ] .

Determinants and inverses of circulant matrices with complex Fibonacci numbers

Ercan Altınışık, N. Feyza Yalçın, Şerife Büyükköse (2015)

Special Matrices

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.

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