Orientadores: Fernando Eduardo Torres Orihuela, Herivelto Martins Borges Filho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T05:54:38Z (GMT). No. of bitstreams: 1 Ribeiro_BeatrizCasularidaMotta_D.pdf: 51476410 bytes, checksum: 46cb0c7a6206a5f0683b23a73ff3938e (MD5) Previous issue date: 2011 / Resumo: Obtemos novos arcos completos associados ao conjunto de pontos racionais de uma certa generalização da curva Hermitiana que é Frobenius não-clássica. A construção está relacionada ao cálculo do número de pontos racionais de uma classe de curvas de Artin-Schreier / Abstract: We obtain new complete arcs arising from the set of rational points of a certain generalization of the Hermitian plane curve which is Frobenius non-classical. Our construction is related to the computation of the number of rational points of a class of Artin-Schreier curves / Doutorado / Matematica / Doutor em Matemática

L'objet de cette thèse est une description effective des espaces de modules des courbes hyper- elliptiques de genre 3 en caractéristiques positives. En caractéristique nulle ou impaire, on obtient une paramétrisation de ces espaces de modules par l'intermédiaire des algèbres d'invariants pour l'action du groupe spécial linéaire sur les espaces de formes binaires de degré 8, qui sont de type fini. Suite aux travaux de Lercier et Ritzenthaler, les cas des corps de caractéristiques 3, 5 et 7 restaient ouverts. Pour ces derniers, les méthodes classiques de la caractéristique nulle sont inno- pérantes pour l'obtention de générateurs pour les algèbres d'invariants en jeu. Nous nous sommes donc contenté d'exhiber des invariants séparants en caractéristiques 3 et 7. En outre, nos résultats concernant la caractéristique 5 suggèrent l'inadéquation de cette approche pour ce cas. À partir de ces résultats, nous avons pu expliciter la stratification des espaces de modules des courbes hyperelliptiques de genre 3 en caractéristiques 3 et 7 selon les groupes d'automorphismes et implémenté divers algorithmes, dont celui de Mestre, pour la reconstruction d'une courbe à partir de son module, ie la valeur de ses invariants. Pour cette phase de reconstruction, nous nous sommes notamment attaché aux questions arithmétiques, comme l'existence d'une obstruction à être un corps de définition pour le corps de module et, dans le cas contraire, à l'obtention d'un modèle de la courbe sur ce corps minimal. Enfin pour la caractéristique 2, notre approche est différente, dans la mesure où les courbes sont étudiées via leur modèle d'Artin-Schreier. Nous exhibons pour celles-ci des invariants bigradués qui dépendent de la structure arithmétique des points de ramifications des courbes. / The aim of this thesis is to provide an explicite description of the moduli spaces of genus 3 hyperelliptic curves in positive characteristic. Over a field of characteristic zero or odd, a parame- terization of these moduli spaces is given via the algebra of invariants of binary forms of degree 8 under the action of the special linear group. After the work of Lercier and Ritzenthaler, the case of fields of characteristic 3, 5 and 7 are still open. However, in these remaining case, the classical methods in characteristic zero do not work in order to provide generators for these algebra of invariants. Hence we provide only separating invariants in characteristic 3 and 7. Furthermore our results in characteristic 5 show this approach is not suitable. From these results, we describe the stratification of the moduli spaces of genus 3 hyperelliptic curves in characteristic 3 and 7 according to the automorphism groups of the curves and imple- ment algorithms to reconstruct a curve from its invariants. For this reconstruction stage, we paid attention to arithmetic issues, like the obstruction to be a field of definition for the field of moduli. Finally, in the characteristic 2 case, we use a different approach, given that the curves are defined by their Artin-Schreier models. The arithmetic structure of the ramification points of these curves stratify the moduli space in 5 cases and we define in each case invariants that characterize the isomorphism class of hyperelliptic curves.

Le problème du logarithme discret est une brique fondamentale de nombreux protocoles de communication sécurisée. Son instantiation sur les courbes elliptiques a permis, grâce à la petite taille des opérandes considérées, le déploiement de primitives asymétriques efficaces sur des systèmes embarqués. De nos jours, les cryptosystèmes utilisant des courbes elliptiques, aussi appelées courbes de genre 1, sont déjà intensément utilisés: il est donc impératif de savoir estimer précisément la robustesse de ces systèmes. L'existence d'attaques mathématiques permettant de transférer un problème de logarithme discret elliptique dans un autre type de courbe algébrique, et la nouvelle compétitivité des courbes de genre 2 imposent de ne pas se restreindre à la seule compréhension du problème sur les courbes elliptiques.Dans cette optique, le sujet de cette thèse se concentre sur les attaques algébriques sur les courbes de genre supérieur à 1. Les algorithmes de type calcul d'indice sont en général préférés pour ce genre d'attaque. Ces algorithmes se déroulent en deux phases: la première, appelée phase de récolte, dispose de nombreuses modélisations algébriques dépendant de la courbe ciblée. Le problème sous-jacent revient à savoir décomposer efficacement des points dans la variété Jacobienne de la courbe en somme d'autres points.On propose dans un premier temps une approche par crible pour la phase de récolte, s'adaptant à tous les types de courbes de genre plus grand que 1, et qui est expérimentalement plus efficaces que les méthodes de l'état de l'art. Cette méthode a l'avantage de proposer une implémentation immédiate, et évite les problèmes usuels de gestion des relations obtenues.Ensuite, on se concentre les variantes de calcul d'indice appelées attaques par décomposition, et ciblant les courbes définies sur des extensions de corps. Dans cette situation, la phase de récolte est effectuée par résolution de nombreux systèmes polynomiaux multivariés. On propose une nouvelle approche de modélisation de ces systèmes, en généralisant la notion de polynômes de sommation elliptique à tout les types de courbes algébriques. Pour cela on fait appel à la théorie de l'élimination, tandis que l'aspect pratique est gérée par des méthodes de bases de Gröbner.Enfin, on fournit des algorithmes d'amélioration du processus de résolution des systèmes lorsque la caractéristique du corps de base est paire. Par le biais d'une présentation théorique générale et en utilisant des méthodes de bases de Gröbner, on propose une analyse fine de l'impact de ces améliorations sur la complexité de la résolution. Cette analyse fine, ainsi qu'une implémentation dédiée, nous permettent d'attaquer une courbe de genre 2 satisfaisant des bornes de sécurité réaliste en pratique. / The discrete logarithm problem is a fundamental brick for several protocols for secured communications. Its instantiation over elliptic curves allows the deployment of efficient asymmetric primitives in embedded systems, because of the small size of the parameters considered. Nowadays, elliptic curves cryptosystems are already widely used: it is therefore crucial to precisely understand the hardness of such systems. Because of the existence of mathematical attacks enabling the transfer from an elliptic curve discrete logarithm problem to another algebraic curve, and the upcoming competitivity of genus 2 curves, it is mandatory to study the problem not only for elliptic curves, but for the other curves as well.In this way, this thesis focuses on the algebraic attacks over curves with genus greater than 1. The index calculus family of algorithms is in general preferred for this kind of attacks. Those algorithms run in two phases: the first, called harvesting phase, can be modelled by different algebraic approaches, depending in the targetted curve. The underlying problem amounts to knowing how to decompose efficiently points in the Jacobian variety of the curve into sums of other points.First, we propose a sieving approach to the harvesting, that can be adated to any type of curves with genus greater than 1, and which turns to be experimentally more efficient than state-of-the-art's methods. Moreover, our method allows a close-to-immediate implementation, and avoid complications coming from relations management.Our second focus is on a variant of index calculus called decomposition attack, targetting curves defined over field extensions. In this situation, harvesting is done by solving numerous multivariate polynomial systems. We propose a new approach to this modelling by generalizing the notion of elliptic summation polynomias to any type of algebraic curves. This uses elimination theory, and the computational aspect is handled by Gröbner bases methods.Third, we give algorithms to improve the solving process of the systems arising from a decomposition attacks when the characteristic of the base field is even. By mean of a general theoretical presentation, and using Gröbner bases methods, we give a sharp analysis of the impact of such improvement on the complexity of the resolution process. This sharp analysis together with a dedicated implementation allows us to attack a genus 2 curve satisfying realistic security parameters.

<p dir="ltr">In this article, we establish results concerning the nilpotent representation theory of groups. In particular, we utilize a theorem of Stallings to provide a general method that constructs pairs of groups that have isomorphic universal nilpotent quotients. We then prove by counterexample that absolute Galois groups of number fields are not determined by their universal nilpotent quotients. We also show that this is the case for residually nilpotent Kleinian groups and in fact, there exist non-isomorphic pairs that have arbitrarily large nilpotent genus. We additionally provide examples of non-isomorphic curves whose geometric fundamental groups have isomorphic universal nilpotent quotients and the isomorphisms are compatible with the outer Galois actions. </p>

Diese Dissertation hat zwei Teile. Im ersten Teil untersuchen wir die Modulräume von Kurven mit multiplen Spinstrukturen. Wir stellen eine neue Kompaktifizierung dieser Räume mit geometrisch sinnvollem Grenzverhalten vor. Die irreduziblen Komponenten dieser Räume werden vollstandig klassifiziert. Die Ergebnisse aus diesem ersten Teil der Dissertation sind fundamental für die Degenerationstechniken im zweiten Teil. Im zweiten Teil untersuchen wir eine Reihe von Problemen, die von der klassischen Geometrie inspiriert werden. Unser Hauptaugenmerk liegt hierbei auf dem Fall von zwei Hyperebenen, die eine kanonische Kurve in jedem Schnittpunkt tangential berühren. Wir fragen, ob eingemensamer Tangentialpunk existieren kann. Unsere Analyse zeigt, dass so ein gemeinsamer Punkt nur in Kodimension 1 im Modulraum existieren kann. Wir berechen dann weiter die Klasse dieses Divisors. Insbesonders zeigen wir, dass diese Klasse eine hinreichend kleine Steigung hat, sodass die kanonischen Klassen von Modulräumen von Kurven mit zwei ungeraden Spinstrukturen gross ist, wenn der Genus grösser ist als neun. Falls die zugehörigen groben Modulräume gutartige Singularitäten haben, dann haben sie in diesem Intervall maximale Kodaria Dimension. / This thesis has two parts. In Part I we consider the moduli spaces of curves with multiple spin structures and provide a compactification using geometrically meaningful limiting objects. We later give a complete classification of the irreducible components of these spaces. The moduli spaces built in this part provide the basis for the degeneration techniques required in the second part. In the second part we consider a series of problems inspired by projective geometry. Given two hyperplanes tangential to a canonical curve at every point of intersection, we ask if there can be a common point of tangency. We show that such a common point can appear only in codimension 1 in moduli and proceed to compute the class of this divisor. We then study the general properties of curves in this divisor. Our divisor class has small enough slope to imply that the canonical class of the moduli space of curves with two odd spin structures is big when the genus is greater than 9. If the corresponding coarse moduli spaces have mild enough singularities, then they have maximal Kodaira dimension in this range.

Diese Dissertation untersucht Brill-Noether-Theorie der algebraischen Kurven, unter besonderer Berücksichtigung von Kurven auf K3-Flächen und Del-Pezzo-Flächen. In Kapitel 2 studieren wir den Gieseker-Petri-Ort GP_g im Modulraum M_g der glatten irreduziblen Kurven vom Geschlecht g. Dieser Ort wird definiert durch Kurven mit einer Brill-Noether-Varietät G^r_d(C), die singulär ist oder deren Dimension größer als erwartet ist. Der Satz von Gieseker-Petri impliziert, dass GP_g mindestens Kodimension 1 hat, und es wurde vermutet, dass er von reiner Kodimension 1 ist. Wir beweisen diese Vermutung für Geschlecht höchstens 13. Dies wird dadurch ermöglicht, dass man für kleine Geschlechter die Dimension der irreduziblen Komponenten von GP_g mittels "ad hoc"-Beweisführungen untersuchen kann. Lazarsfelds Beweis des Gieseker-Petri-Theorems mittels Kurven auf allgemeninen K3-Flächen deutet darauf hin, dass die Brill-Noether-Theorie von K3-Schnitten wichtig ist, um den Gieseker-Petri-Ort besser zu verstehen. Linearscharen von Kurven, die auf K3-Flächen liegen, stehen in tiefgehender Beziehung zu sogenannten Lazarsfeld-Mukai-Vektorbündeln. In Kapitel 3 untersuchen wir die Stabilität der Lazarsfeld-Mukai-Vektorbündel vom Rang 3 auf einer K3-Fläche S, und wir zeigen, dass sie viele Informationen über Netze vom Typ g^2_d auf Kurven in S enthalten. Wenn d größ genug ist, erhalten wir eine obere Schranke für die Dimension der Varietät G^2_d(C). Wenn die Brill-Noether-Zahl negativ ist, beweisen wir, dass jedes g^2_d in einer von einem Geradenbündel induzierten Linearschar enthalten ist, wie von Donagi und Morrison vermutet wurde. Kapitel 4 befasst sich mit Syzygien einer gegebenen Kurve C, die auf einer Del-Pezzo-Fläche liegt. Wir insbesondere, dass C die Greens Vermutung erfüllt, die impliziert, dass die Existenz gewisser spezieller Linearscharen auf C von den Gleichungen ihrer kanonischen Einbettung abgelesen werden kann. / We investigate Brill-Noether theory of algebraic curves, with special emphasis on curves lying on $K3$ surfaces and Del Pezzo surfaces. In Chapter 2, we study the Gieseker-Petri locus GP_g inside the moduli space M_g of smooth, irreducible curves of genus g. This consists, by definition, of curves [C] in M_g such that for some r, d the Brill-Noether variety G^r_d(C), which parametrizes linear series of type g^r_d on C, either is singular or has some components exceeding the expected dimension. The Gieseker-Petri Theorem implies that GP_g has codimension at least 1 in M_g and it has been conjectured that it has pure codimension 1. We prove this conjecture up to genus 13; this is possible since, when the genus is low enough, one is able to determine the irreducible components of GP_g and to study their codimension by "ad hoc" arguments. Lazarsfeld''s proof of the Gieseker-Petri-Theorem by specialization to curves lying on general K3 surfaces suggests the importance of the Brill-Noether theory of K3-sections for a better understanding of the Gieseker-Petri locus. Linear series on curves lying on a K3 surface are deeply related to the so-called Lazarsfeld-Mukai bundles. In Chapter 3, we study the stability of rank-3 Lazarsfeld-Mukai bundles on a K3 surface S, and show it encodes much information about nets of type g^2_d on curves C contained in S. When d is large enough and C is general in its linear system, we obtain a dimensional statement for the variety G^2_d(C). If the Brill-Noether number is negative, we prove that any g^2_d is contained in a linear series which is induced from a line bundle on S, as conjectured by Donagi and Morrison. Chapter 4 concerns syzygies of any given curve C lying on a Del Pezzo surface S. In particular, we prove that C satisfies Green''s Conjecture, which implies that the existence of some special linear series on C can be read off the equations of its canonical embedding.

Diese Arbeit widmet sich der Untersuchung von zwei Problemen der abzählenden Geometrie im Zusammenhang mit linearen Systemen auf algebraischen Kurven. Das erste Problem besteht darin, die Frage der Gültigkeit der Jonquières-Formeln zu klären. Diese Formeln berechnen die Anzahl von Divisoren mit vorgeschriebener Multiplizität, genannt de Jonquières-Divisoren, die in einem linearen System auf einer glatten projektiven Kurve enthalten sind. Um dies zu tun, konstruieren wir den Raum der de Jonquières-Divisoren als einen Determinantenzyklus des symmetrischen Produkts der Kurve und beweisen, dass er für eine allgemeine Kurve die erwartete Dimension hat. Dabei beschreiben wir die Degenerationen der Jonquières-Divisoren zu den Knotenkurven sowohl mit linearen Systemen als auch mit kompaktifizierten Picard-Schemata. Das zweite Problem behandelt Zyklen von Untergeordneten-, oder allgemeiner, Sekanten-Divisoren zu einem gegebenen linearen System auf einer Kurve. Wir betrachten den Durchschnitt zweier solcher Zyklen, die Sekanten-Divisoren von zwei verschiedenen linearen Systemen auf der gleichen Kurve entsprechen, und untersuchen die Gültigkeit der enumerativen Formeln, die die Anzahl der Teiler im Durchschnitt zählen. Wir untersuchen einige interessante Fälle mit unerwarteten Transversalitätseigenschaften und etablieren eine allgemeine Methode, um zu überprüfen, wann dieser Durchschnitt leer ist. / This thesis is dedicated to the study of two enumerative geometry problems in the context of linear series on algebraic curves. The first problem is that of settling the issue of the validity of the de Jonquières formulas. These formulas compute the number of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. To do so, we construct the space of de Jonquières divisors as a determinantal cycle of the symmetric product of the curve and prove that, for a general curve with a general linear series, it is of expected dimension. In doing so, we describe the degenerations of de Jonquières divisors to nodal curves using both limit linear series and compactified Picard schemes. The second problem deals with cycles of subordinate or, more generally, secant divisors to a given linear series on a curve. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

La parution en 1750 de l'Introduction à l'analyse des lignes courbes algébriques, unique ouvrage publié de Gabriel Cramer, professeur de mathématiques à l'Académie de Genève, est un jalon important dans l'histoire de la géométrie des courbes et de l'algèbre. Il s'est passé près de dix années entre le moment où le Genevois a écrit les premières lignes de son traité des courbes, à l'automne 1740, et sa publication effective : il s'agit de l'œuvre d'une vie.Ce livre a une histoire, à la fois intellectuelle et matérielle, qui s'inscrit dans les contextes scientifiques, professionnels, académiques et sociaux dans lesquels évoluent son auteur puis ses lecteurs. De la genèse d'un texte manuscrit en devenir dont nous suivrons les évolutionsau cours du processus d'écriture et au gré des événements de la vie de son auteur, aux différentes lectures mathématiciennes et historiennes du texte publié qui en sont faites dans les quelque deux siècles qui ont suivi sa publication, je souhaite ici écrire, pour autant que cette expression ait un sens - ce que je m'attacherai à démontrer - une « biographie » de l'Introduction de Gabriel Cramer / The publication in 1750 of the Introduction à l'analyse des lignes courbes algébriques, the only published work by Gabriel Cramer, professor of mathematics at the Geneva Academy, is an important milestone in the history of geometry of curves and algebra. Almost ten years passed between the time when the Genevan wrote the first lines of his treatise on curves in the autumn of 1740 and its actual publication, making it a lifetime achievement.This book has a history, both intellectual and material, which takes place in the scientific, professional, academic and social contexts in which evolve its author and its readers. From the genesis of a handwritten text as a work in progress of which we will follow the evolutions during the process of writing and according to the events of its author's life, to the various mathematicians and historians' readings of the published text which are made in the two centuries following its publication, I would like to write here, insofar as this expression makes sense - which I shall endeavour to demonstrate - a « biography » of Gabriel Cramer's Introduction

Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T19:35:36Z (GMT). No. of bitstreams: 1 Negreiros_DiogoBrunoFernandes_M.pdf: 5674415 bytes, checksum: bdd28225d3cc5505f91fd61e797f2794 (MD5) Previous issue date: 2011 / Resumo: Este texto tem como objetivo o estudo de um tipo de código que possui relações com as teorias de curvas algébricas e de formas quadráticas. Começaremos introduzindo as definições e resultados sobre as três teorias que serão necessárias a este estudo. Depois apresentaremos os códigos a serem estudados bem como as relações entre seus sub-códigos e curvas algébricas e entre suas palavras e formas quadráticas. Observando que sub-códigos de peso mais baixo correspondem a curvas com mais pontos, nos dedicaremos a obter um processo para a descoberta de sub-códigos de peso mínimo dentro deste tipo de código. Tal processo será possível através de investigações sobre as formas quadráticas associadas a palavras. Finalizaremos com exemplos de aplicações do processo em alguns códigos, o que permite também calcular seus pesos de Hamming generalizados de ordem mais baixa / Abstract: This text's objective is the study of a kind of code wich has relations with the theories of algebraic curves and quadratic forms. We start by introducing definitions and results about the three theories we will need in such study. Later, we present the codes wich will be studied along with relations between its subcodes and algebraic curves and between its words and quadratic forms. Noting that lower weight subcodes correspond to curves with more points, we research a process to find minimum weight subcodes in this kind of code. This process will be possible through investigations on the quadratic forms related to words. Finally we set examples of applications of the process on some codes, and that gives us their lower order generalized Hamming weights / Mestrado / Matematica / Mestre em Matemática

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.