Regular Choice Functions and Uniformisations For countable Domains

We view languages of words over a product alphabet A x B as relations between words over A and words over B. This leads to the notion of regular relations - relations given by a regular language. We ask when it is possible to find regular uniformisations of regular relations. The answer depends on the structure or shape of the underlying model: it is true e.g. for ?-words, while false for words over ? or for infinite trees. In this paper we focus on countable orders. Our main result characterises, which countable linear orders D have the property that every regular relation between words over D has a regular uniformisation. As it turns out, the only obstacle for uniformisability is the one displayed in the case of ? - non-trivial automorphisms of the given structure. Thus, we show that either all regular relations over D have regular uniformisations, or there is a non-trivial automorphism of D and even the simple relation of choice cannot be uniformised. Moreover, this dichotomy is effective

Images of Galois representations and p-adic models of Shimura curves

The thesis treats two questions situated in the Langlands program, which is one of the most active and important areas in current number theory and arithmetic geometry. The first question concerns the study of images of Galois representations into Hecke algebras coming from modular forms over finite fields, and the second one deals with p-adic models of Shimura curves and its bad reduction. Consequently, the thesis is divided in two parts. The first part is concerned with the study of images of Galois representations that take values in Hecke algebras of modular forms over finite fields. The main result of this part is a complete classification of the possible images of 2-dimensional Galois representations with coefficients in local algebras over finite fields under the hypotheses that: (i) the square of the maximal ideal is zero, (ii) that the residual image is big (in a precise sense), and (iii) that the coefficient ring is generated by the traces. In odd characteristic, the image is completely determined by these conditions; in even characteristic the classification is much richer. In this case, the image is uniquely determined by the number of different traces of the representation, a number which is given by an easy formula. As an application of these results, the existence of certain p-elementary abelian extensions of big non-solvable number fields can be deduced. Whereas some aspects of class field theory are accessible through this approach, it can be applied to huge fields for which standard techniques totally fail. The second part of the thesis consists of an approach to p-adic uniformisations of Shimura curves X(Dp,N) through a combination of different techniques concerning rigid analytic geometry and arithmetic of quaternion orders. The results in this direction lean on two methods: one is based on the information provided by certain Mumford curves covering Shimura curves and the second one on the study of Eichler orders of level N in the definite quaternion algebra of discriminant D. Combining these methods, an explicit description of fundamental domains associated to p-adic uniformisation of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D,N) is 1, is given. The method presented in this thesis enables one to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains and their stable reduction-graphs. As an application, general formulas for the reduction-graphs with lengths at p of the considered families of Shimura curves can be computed

On the synthesis of integral and dynamic recurrences

PhD ThesisSynthesis techniques for regular arrays provide a disciplined and well-founded approach to the design of classes of parallel algorithms. The design process is guided by a methodology which is based upon a formal notation and transformations. The mathematical model underlying synthesis techniques is that of affine Euclidean geometry with embedded lattice spaces. Because of this model, computationally powerful methods are provided as an effective way of engineering regular arrays. However, at present the applicability of such methods is limited to so-called affine problems. The work presented in this thesis aims at widening the applicability of standard synthesis methods to more general classes of problems. The major contributions of this thesis are the characterisation of classes of integral and dynamic problems, and the provision of techniques for their systematic treatment within the framework of established synthesis methods. The basic idea is the transformation of the initial algorithm specification into a specification with data dependencies of increased regularity, so that corresponding regular arrays can be obtained by a direct application of the standard mapping techniques. We will complement the formal development of the techniques with the illustration of a number of case studies from the literature.EPSR