Cette thèse traite du comptage d'extensions cubiques relatives. Dans le premier chapitre on traite un travail commun avec Henri Cohen. Soit k un corps de nombres. On donne une formule asymptotique pour le nombre de classes d'isomorphisme d'extensions cubiques L/k telles que la clôture galoisienne de L/k contienne une extension quadratique fixée K_2/k. L'outil principal est la théorie de Kummer. Dans le second chapitre, on suppose k un corps quadratique imaginaire (avec nombre de classes 1) et on décrit un algorithme pour énumérer toutes les classes d'isomorphisme d'extensions cubiques L/k jusqu'à une certaine borne X sur la norme du discriminant relatif. / This thesis is about counting relative cubic extensions. In the first chapter we describe a joint work with Henri Cohen. Let k be a number field. We give an asymptotic formula for the number of isomorphism classes of cubic extensions L/k such that the Galois closure of L/k contains a fixed quadratic extension K_2/k. The main tool is Kummer theory. In the second chapter, we suppose k to be an imaginary quadratic number field (with class number 1) and we describe an algorithm for listing all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant ideal.

Les courbes algébriques sont des objets centraux de la géométrie algébrique. Dans cette thèse, nous étudions ces objets sous différents angles de la géométrie algébrique tels que la géométrie algébrique effective et la géométrie arithmétique. Dans le premier chapitre, nous étudions les courbes non-hyperelliptiques de genre g et leurs jacobiennes liées par l’intermédiaire de diviseurs thêta caractéristiques. Ces derniers contiennent des propriétés géométriques extrinsèques qui permettent de calculer les constantes thêta. Dans le deuxième chapitre, nous nous concentrons sur les courbes hyperelliptiques de genre 2 et leur surface de Kummer associée avec une motivation cryptographique. Dans le troisième et dernier chapitre, nous étudions les revêtements doubles non-ramifiés des courbes non-hyperelliptiques de genre g pour obtenir des informations sur le p-rang. / Algebraic curves are central objects in algebraic geometry. In this thesis, we consider these objects from different angles of algebraic geometry such as computational algebraic geometry and arithmetic geometry. In the first chapter, we study non-hyperelliptic curves of genus g and their Jacobians linked via theta characteristic divisors. Such divisors provide extrinsic geometric properties which allow us to compute theta constants. In the second chapter, we focus on hyperelliptic curves of genus 2 and the associated Kummer surface with a cryptographic motivation. In the third and final chapter, we examine unramified double covers of non-hyperelliptic curves of genus g to obtain information about p-rank.

Until the nineteenth century, the violoncello was considered a background accompaniment instrument. By 1900 however, over eighty method books had been published for cello, and Richard Wagner and Richard Strauss were composing orchestral cello parts equal in difficulty to those of the violin, traditionally the only virtuosic string part. The emancipation from the ties of bass ostinato for the cello began with Bernhard Romberg in Dresden. The group of cellists, who came to be known as the Dresden School, included Kummer, Lee, Goltermann, Cossmann, Popper, Grutzmacher, Davidov, and other cellists that were students and colleagues of this group. The Dresden School of cellists attempted not only to bring the instrument into prominence, but to revolutionize the technique of the instrument completely. The cello pedagogues of the Dresden School achieved this by publishing their methods and advancements in technique in cello etude and method books. This efficient process of dissemination allowed for the members of the school to improve on each other's work over time. By the second half of the nineteenth century, the cello pedagogy of the Dresden School was established through the etudes published by the cellist-composers of the Dresden School, and these etudes are still considered some of the most advanced studies for cello, and are the foundation of modern cello pedagogy. At the turn of the twentieth century the Dresden School was the leading cello school in the world, and no longer tied only to the city of Dresden, but spread throughout Europe and beyond. In the publishing of their etudes, the Dresden cellists not only passed down their information to their students, but also to future generations of cellists. Descendants of the Dresden School cellists are now performing in almost every nation and teaching the ideas born in nineteenth century Germany.

In dieser Dissertation konstruieren wir neue Beispiele von koassoziativen Untermannigfaltigkeiten und G2-Instantonen in kompakten G2-Mannigfaltigkeiten, die aus Joyces verallgemeinerter Kummer Konstruktion hervorgehen. Die besondere Eigenschaft der in dieser Arbeit gefundenen koassoziativen Untermannigfaltigkeiten ist, dass ihr (topologisch bestimmtes) Volumen gegen Null geht, wenn die umgebende Mannigfaltigkeit sich ihrem Orbifaltigkeits-Limes annähert. Dies ist im Sinne eines Vorschlags von Halverson und Morrison, der darauf hinweist, dass bestimmte Entartungen (oder, allgemeiner, die Perioden) von G2-Strukturen durch das Verhalten von G2-topologischen Größen wie dem Volumen von assoziativen und koassoziativen Untermannigfaltigkeiten nachweisbar sein könnten. Die Konstruktion dieser koassoziativen Untermannigfaltigkeiten ist Inhalt von Kapitel 3 und basiert auf der Deformation von „Modell-Untermannigfaltigkeiten“. Diese Untermannigfaltigkeiten liegen innerhalb des kritischen Bereiches der umgebenden Mannigfaltigkeit, in welchem die Metrik entartet. Abschnitt 3.3 beinhaltet zahlreiche Beispiele von koassoziativen Untermannigfaltigkeiten, die wir durch diese Methode konstruieren. Des Weiteren beschreiben wir die Deformationsfamilie dieser koassoziativen Untermannigfaltigkeiten. In Kapitel 4 konstruieren wir neue Beispiele von G2-Instantonen über verallgemeinerten Kummer Konstruktionen. Wir konzentrieren uns hierbei hauptsächlich auf Auflösungen von Orbifaltigkeiten, deren singuläre Strata von Kodimension 6 sind. Wie im vorherigen Kapitel basiert die Konstruktion dieser Instantonen auf einem Klebesatz, welcher einen Zusammenhang deformiert, der (im quantifizierten Sinne) fast ein G2-Instanton ist. Außerdem benutzen wir Gruppenwirkungen um die Obstruktionen zu reduzieren. Mithilfe dieser Methode konstruieren wir in Abschnitt 4.4 eine unendliche Familie von G2-Instantonen auf einem Bündel über einer bestimmten Kummer Konstruktion. / In this thesis we construct new examples of coassociative submanifolds and G2-instantons in compact G2-manifolds arising from Joyce’s generalised Kummer construction. The special feature of the coassociative submanifolds found in this thesis is that their (topologically determined) volume shrinks to zero as the ambient manifold approaches its orbifold limit. This is in the spirit of a proposal by Halverson and Morrison which indicates that certain degenerations (or, more general, the periods) of G2-structures may be detectable by the behaviour of G2-topological quantities such as the volume of associative and coassociative submanifolds. The construction of these coassociative submanifolds is the content of Chapter 3. It is based on the deformation of ‘model-submanfiolds’. These submanifolds lie within the critical locus of the ambient manifold in which the metric degenerates. Section 3.3 contains numerous examples of coassociative submanifolds which we construct via this method. Furthermore, we give a description of the deformation family of these coassociative submanifolds. In Chapter 4 we construct new examples of G2-instantons over generalised Kummer constructions. We focus mainly on resolutions of orbifolds whose singular strata are of codimension 6. As in the previous chapter, the construction of these instantons is based on a gluing theorem which deforms a connection that is (in a quantified sense) close to being a G2-instanton. Furthermore, we use group actions to reduce the obstructions. Using this method, we construct in Section 4.4 an infinite family of G2-instantons on a bundle over one particular Kummer construction.

Dans ce mémoire, on s'intéresse à des briques utiles à la cryptographie asymétrique et principalement au problème du logarithme discret. Dans une première partie, nous présentons un survol de différentes notions algorithmiques de couplages sur des jacobiennes de courbes de genre 2 et décrivons les détails d'une implémentation soigneuse. Nous faisons une comparaison à niveau de sécurité équivalent avec les couplages sur les courbes elliptiques. Une deuxième partie est dévolue à la recherche de modèles efficaces pour les courbes elliptiques et les surfaces de Kummer non-ordinaires en caractéristique 2. Pour le genre 1, nous obtenons que le modèle d'Edwards binaire se déduit du modèle d'Edwards classique en caractéristique zéro. Pour le genre $2$, nous utilisons des techniques de "déformation" qui consistent à considérer une famille de jacobiennes sur un anneau des séries formelles, telle que la fibre générique soit ordinaire et la fibre spéciale soit la jacobienne considérée. Il s'agit alors de montrer que la loi de groupe sur la fibre générique s'étend à tout le modèle. Nous comparons les lois de composition ainsi obtenues avec celles déjà connues.

Inaug.-Diss.--Cologne. / Vita. Bibliography: p. [146]-147.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22981号 / 理博第4658号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 小野 薫, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

<p>This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of N_q(g) is now known.</p><p>At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.</p><p> </p>

This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

L'étude du nombre de points rationnels d'une courbe définie sur un corps fini se divise naturellement en deux cas : lorsque le genre est petit (typiquement g<=50), et lorsqu'il tend vers l'infini. Nous consacrons une partie de cette thèse à chacun de ces cas. Dans la première partie de notre étude nous expliquons comment calculer l'équation de n'importe quel revêtement abélien d'une courbe définie sur un corps fini. Nous utilisons pour cela la théorie explicite du corps de classe fournie par les extensions de Kummer et d'Artin-Schreier-Witt. Nous détaillons également un algorithme pour la recherche de bonnes courbes, dont l'implémentation fournit de nouveaux records de nombre de points sur les corps finis d'ordres 2 et 3. Nous étudions dans la seconde partie une formule de trace d'opérateurs de Hecke sur des formes modulaires quaternioniques, et montrons que les courbes de Shimura associées forment naturellement des suites récursives de courbes asymptotiquement optimales sur une extension quadratique du corps de base. Nous prouvons également qu'alors la contribution essentielle en points rationnels est fournie par les points supersinguliers. / The study of the number of rational points of a curve defined over a finite field naturally falls into two cases: when the genus is small (typically g<=50), and when it tends to infinity. We devote one part of this thesis to each of these cases. In the first part of our study, we explain how to compute the equation of any abelian covering of a curve defined over a finite field. For this we use explicit class field theory provided by Kummer and Artin-Schreier-Witt extensions. We also detail an algorithm for the search of good curves, whose implementation provides new records of number of points over the finite fields of order 2 and 3. In the second part, we study a trace formula of Hecke operators on quaternionic modular forms, and we show that the associated Shimura curves of the form naturally form recursive sequences of asymptotically optimal curves over a quadratic extension of the base field. Moreover, we then prove that the essential contribution to the rational points is provided by supersingular points.