Arithmetic, Geometry and Coding Theory (AGCT 2003) by Yves Aubry, Gilles Lachaud PDF

By Yves Aubry, Gilles Lachaud

Résumé :
Arithmétique, géométrie et théorie des codes (AGCT 2003)
En mai 2003 se sont tenus au Centre overseas de Rencontres Mathématiques à Marseille (France), deux événements centrés sur l'Arithmétique, l. a. Géométrie et leurs functions à los angeles théorie des Codes ainsi qu'à l. a. Cryptographie : une école Européenne ``Géométrie Algébrique et Théorie de l'Information'' ainsi que los angeles 9ème édition du colloque foreign ``Arithmétique, Géométrie et Théorie des Codes''. Certains des cours et des conférences font l'objet d'un article publié dans ce quantity. Les thèmes abordés furent à los angeles fois théoriques pour certains et tournés vers des functions pour d'autres : variétés abéliennes, corps de fonctions et courbes sur les corps finis, groupes de Galois de pro-p-extensions, fonctions zêta de Dedekind de corps de nombres, semi-groupes numériques, nombres de Waring, complexité bilinéaire de l. a. multiplication dans les corps finis et problèmes de nombre de classes.

Mots clefs : Fonctions zêta, variétés abéliennes, corps de fonctions, courbes sur les corps finis, excursions de corps de fonctions, corps finis, graphes, semi-groupes numériques, polynômes sur les corps finis, cryptographie, courbes hyperelliptiques, représentations p-adiques, excursions de corps de classe, groupe de Galois, issues rationels, fractions maintains, régulateurs, nombre de sessions d'idéaux, complexité bilinéaire, jacobienne hyperelliptiques

Abstract:
In could 2003, occasions were held within the ``Centre foreign de Rencontres Mathématiques'' in Marseille (France), dedicated to mathematics, Geometry and their purposes in Coding thought and Cryptography: an ecu college ``Algebraic Geometry and data Theory'' and the 9-th overseas convention ``Arithmetic, Geometry and Coding Theory''. many of the classes and the meetings are released during this quantity. the themes have been theoretical for a few ones and became in the direction of functions for others: abelian kinds, functionality fields and curves over finite fields, Galois staff of pro-p-extensions, Dedekind zeta features of quantity fields, numerical semigroups, Waring numbers, bilinear complexity of the multiplication in finite fields and sophistication quantity problems.

Key phrases: Zeta features, abelian kinds, capabilities fields, curves over finite fields, towers of functionality fields, finite fields, graphs, numerical semigroups, polynomials over finite fields, cryptography, hyperelliptic curves, p-adic representations, classification box towers, Galois teams, rational issues, endured fractions, regulators, excellent type quantity, bilinear complexity, hyperelliptic jacobians

Class. math. : 14H05, 14G05, 11G20, 20M99, 94B27, 11T06, 11T71, 11R37, 14G10, 14G15, 11R58, 11A55, 11R42, 11Yxx, 12E20, 14H40, 14K05

Table of Contents

* P. Beelen, A. Garcia, and H. Stichtenoth -- On towers of functionality fields over finite fields
* M. Bras-Amorós -- Addition habit of a numerical semigroup
* O. Moreno and F. N. Castro -- at the calculation and estimation of Waring quantity for finite fields
* G. Frey and T. Lange -- Mathematical historical past of Public Key Cryptography
* A. Garcia -- On curves over finite fields
* F. Hajir -- Tame pro-p Galois teams: A survey of modern work
* E. W. Howe, okay. E. Lauter, and J. most sensible -- unnecessary curves of genus 3 and four
* D. Le Brigand -- genuine quadratic extensions of the rational functionality box in attribute two
* S. R. Louboutin -- specific top bounds for the residues at s=1 of the Dedekind zeta services of a few absolutely actual quantity fields
* S. Ballet and R. Rolland -- at the bilindar complexity of the multiplication in finite fields
* Yu. G. Zarhin -- Homomorphisms of abelian forms

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10 this implies t(F ) = ν(F ). Moreover, we have seen that the completely splitting places in the tower F are described by solutions of the functional equation for ϕ(t) := (1−t)/tq and ψ(t) := (tq +t−1)/t. 4 that ´ ` 11 SEMINAIRES & CONGRES TOWERS OF FUNCTION FIELDS 19 essentially only one solution H(t, s) exists, we would be done. , Pω is defined as the zero of x1 − ω) would then be given by H(ω q + ω − 1, ω) = 0. As it is, we cannot apply the proposition directly. However, we can rewrite the defining equation of the tower F .

Math. (to appear). G. G. Vladut – The number of points of an algebraic curve, Func. Anal. 17 (1983), p. 53–54. M. Duursma, B. Poonen & M. L. Mullen, A. Poli & H. ), Lecture Notes in Computer Science, vol. 2948, Springer, 2004, p. 148–153. [9] A. Garcia & H. Stichtenoth – On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 61 (1996), p. 248–273. , Skew pyramids of function fields are asymptotically bad, in Coding Theory, [10] Cryptography and Related Areas (J.

A numerical semigroup Λ is uniquely determined by the binary operation ⊕. Proof. — We will show that Λ is unique by proving that λi is uniquely determined by ⊕ for all i ∈ N0 . 4, i ⊕ j j + λi for all j, i ⊕ j = j + λi for all j with λj c. Therefore, maxj {i ⊕ j − j} exists for all i, is uniquely determined by ⊕ and it is exactly λi . 2. The sequence (νi ) determines a semigroup In this section we prove that any numerical semigroup is uniquely determined by the associated sequence (νi ). We will use the following well-known result on the values νi .

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