Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n &gt / 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group M of N.

Cette thèse porte sur l'étude de certaines familles de représentations projectives des groupes modulaires de surfaces issues de théories topologiques quantiques de champs. Les résultats principaux portent sur leur décomposition en facteurs irréductibles. / This thesis deals with some families of projective representations of the mapping class groups of surfaces arising in topological quantum field theories. The main results concern their decomposition into irreducible factors.

Dans cette thèse nous étudions certains propriétés combinatoires et géométriques des complexes d'arcs des surfaces de type fini. Nous démontrons que le groupe d'automorphisme du complexe d'arcs est le mapping class group de la surface. Nous étudions aussi le graphe des triangulations idéales et nous donnons certains applications au espaces de Teichmueller des surfaces avec bord . / In this thesis we deal with combinatorial and geometric properties of the arc complex of a surface of finite type. We prove that its automorphism group is isomorphic to the mapping class of the surface. Furthermore, we investigate the geometric properties of the ideal triangulation graph of a surface and provide some application to Teichmueller theory of a surface with boundary .

Let Q(√(-d)) be an imaginary quadratic field with discriminant Δ. We use the isomorphism between the ideal class groups of the field and the equivalence classes of binary quadratic forms to find the structure of the class group. We determine the structure by combining two of Shanks' algorithms [7, 8]. We utilize this method to find fields with cyclic factors that have order a large power of 2, or fields with class groups of high 5-ranks or high 7-ranks. / Master of Science

Title: Solving diophantine equations by factorization in number fields Author: Bc. Maroš Hrnčiar Department: Department of Algebra Supervisor: Mgr. Vítězslav Kala, Ph.D., Mathematical Institute, University of Göttingen Abstract: The question of solvability of diophantine equations is one of the oldest mathematical problems in the history of mankind. While different approaches have been developed for solving certain types of equations, this thesis predo- minantly deals with the method of factorization over algebraic number fields. The idea behind this method is to express the equation in the form L = yn where L equals a product of typically linear factors with coefficients in a particular number field. Provided that several assumptions are met, it follows that each of the factors must be the n-th power of an element of the field. The structure of number fields plays a key role in the application of this method, hence a crucial part of the thesis presents an overview of algebraic number theory. In addition to the general theoretical part, the thesis contains all the necessary computations in specific quadratic and cubic number fields describing their basic characteristics. However, the main objective of this thesis is solving specific examples of equati- ons. For instance, in the case of equation x2 + y2 = z3 we...

Dans le premier chapitre de ce travail, nous rappelons la théorie des représentants efficaces d'un élément pseudo-Anosov du mapping class group d'une surface S compacte orientée munie de n+1 points marqués. Ces objets ont été introduits par Bestvina-Handel et Los.<br /><br />Le deuxième chapitre contient l'exposé de la théorie des bons représentants et des représentants super efficaces d'un homéomorphisme pseudo-Anosov f fixant le point marqué x_0. Nous montrons ensuite un résultat de structure sur l'ensemble des représentants super efficaces : cet ensemble est une union d'un nombre fini de cycles qui sont parcourus en appliquant des opérations combinatoires. Nous en déduisons des algorithmes permettant de décider si l'homéomorphisme f - ou, ce qui est équivalent, sa classe d'isotopie - admet une racine fixant x_0, ou commute avec un élément d'ordre fini fixant x_0. Nous en déduisons également une nouvelle solution au problème de conjugaison parmi les éléments pseudo-Anosov du mapping class group qui fixent x_0.<br /><br />Dans le troisième chapitre, nous considérons un homéomorphisme f du disque et O une orbite de période n>=3 pour f. Nous donnons une minoration de l'entropie topologique des homéomorphismes isotopes à f relativement à O. Cette minoration est obtenue à l'aide de la théorie des représentants efficaces.<br /><br />Dans le quatrième chapitre, nous donnons des conditions nécessaires et suffisantes pour qu'une tresse beta à n brins admette une déstabilisation ou un mouvement d'échange. Ces conditions sont des propriétés sur l'élément du mapping class group induit par la tresse beta.

There are several reasons to be interested in conformal field theories in two dimensions. Apart from arising in various physical applications, ranging from statistical mechanics to string theory, conformal field theory is a class of quantum field theories that is interesting on its own. First of all there is a large amount of symmetries. In addition, many of the interesting theories satisfy a finiteness condition, that together with the symmetries allows for a fully non-perturbative treatment, and even for a complete solution in a mathematically rigorous manner. One of the crucial tools which make such a treatment possible is provided by category theory. This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group ofthe Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theories whose chiral data are described by the category of left-modules over H.

The thesis is concerned with the theory of binary quadratic forms with coefficients in the ring of algebraic integers of a number field. Under the assumption that the number field is of narrow class number one, there is developed a theory of composition of such quadratic forms. For a given discriminant, the composition is determined by a bijection between classes of quadratic forms and a so-called relative oriented class group (a group closely related to the class group). Furthermore, Bhargava cubes are generalized to cubes with entries from the ring of algebraic integers; by using the composition of quadratic forms, the composition of Bhargava cubes is proved in the generalized case. 1