In this thesis integral points on affine del Pezzo surfaces are studied. The first two chapters offer a review of arithmetic techniques and del Pezzo surfaces, but also a novel approach to del... Show moreIn this thesis integral points on affine del Pezzo surfaces are studied. The first two chapters offer a review of arithmetic techniques and del Pezzo surfaces, but also a novel approach to del Pezzo surfaces using a new type of surface, namely the peculiar del Pezzo surface. This allows one to study del Pezzo surfaces by considering linear subsystems of cubic plane curves.In Chapter 3 a uniform bound is given for the Brauer group of certain affine del Pezzo surfaces over number fields. This answers an open question for the geometrically related K3 surfaces. While determining this bound techniques are described for (partially) computing these groups.Chapter 4 begins with constructing models of del Pezzo surfaces; not by geometrically manipulating the projective plane over the rationals and taking the flat closure over the integers, but by manipulating schemes over the integers. The advantage of this approach is that one can control the reduction of the surface over all primes. Using these techniques and the computations from Chapter 3 we describe families of surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle. These stand out against previously published examples, since these were all of lower order. Show less
This work is dedicated to interpreting in cohomological terms the special values of zeta functions of arithmetic schemes. Baptiste Morin and Matthias Flach gave a construction of Weil-étale... Show moreThis work is dedicated to interpreting in cohomological terms the special values of zeta functions of arithmetic schemes. Baptiste Morin and Matthias Flach gave a construction of Weil-étale cohomology using Bloch's cycle complexes and stated a precise conjecture for the special values of proper regular arithmetic schemes at any integer argument s=n. The goal of this thesis is to generalize their constructions to arbitrary arithmetic schemes (possibly singular or non-proper), while restricting to the case n < 0. We prove that the resulting conjecture is compatible with the decomposition of an arbitrary scheme into an open subscheme and its closed complement. We also show that this conjecture for an arithmetic scheme X at s=n is equivalent to the conjecture for A^r_X at s=n-r, for any r >= 0. It follows that, taking as an input the schemes for which the conjecture is known, it is possible to construct new schemes, possibly singular or non-proper, for which the conjecture holds as well. This is the main unconditional outcome of the machinery developed in this thesis. Show less
This thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are... Show moreThis thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are examples of arithmetic surfaces. Therefore, Arakelov theory (intersection theory on arithmetic surfaces) occupies a prominent place in this thesis. Apart from this, a substantial part of it is devoted to studying modular curves over finite fields, and their Jacobian varieties, from an algorithmic viewpoint. The end product of this thesis is an algorithm for computing modular Galois representations. These are certain two-dimensional representations of the absolute Galois group of the rational numbers that are attached to Hecke eigenforms over finite fields. The running time of our algorithm is (under minor restrictions) polynomial in the length of the input. This main result is a generalisation of that of work of Jean-Marc Couveignes, Bas Edixhoven et al. Several intermediate results are developed in sufficient generality to make them of interest to the study of modular curves and modular forms in a wider sense. Show less
