In this thesis, rational points on K3 surfaces are studied. In the first part of Chapter 1 the Brauer group and the the Brauer-Manin obstruction are introduced. The second part of Chapter 1 is... Show moreIn this thesis, rational points on K3 surfaces are studied. In the first part of Chapter 1 the Brauer group and the the Brauer-Manin obstruction are introduced. The second part of Chapter 1 is devoted to the introduction of the refined Swan conductor and the link between the latter and the Brauer-Manin set. The aim of Chapter 2 is to give the first example of a K3 surface over a number field (in fact, over the rational numbers) for which a prime of good reduction is involved in the Brauer-Manin obstruction; this gives a negative answer to a question originally asked by Swinnerton-Dyer. In Chapter 3 the role of primes of good reduction in the Brauer-Manin obstruction is discussed, with a focus on K3 surfaces. Finally, in the last chapter, after introducing some techniques that can be used to compute the refined Swan conductor, several examples of Brauer-Manin obstruction on K3 surfaces are provided, with a focus on Kummer K3 surfaces. Show less
This thesis studies the geometry of representation varieties and character stacks. These are spaces parametrizing the representations of a finitely generated group, typically the fundamental group... Show moreThis thesis studies the geometry of representation varieties and character stacks. These are spaces parametrizing the representations of a finitely generated group, typically the fundamental group of a compact manifold, into an algebraic group G. We provide a number of methods to study these spaces through their invariants, focussing on the invariant that takes values in the Grothendieck ring of varieties. We show how both the arithmetic method, which studies the character stacks of compact orientable surfaces through counting points over finite fields, and the geometric method, which studies these character stacks using stratifications, can be expressed in terms of topological quantum field theories. We compute explicitly the invariants in specific cases, such as for G = SL2 and for G equal to the groups of upper triangular matrices. Motivated by these applications, we develop a number of new computational tools. Finally, we study the representation varieties and character stacks of the free groups and of the free abelian groups, of finite rank. We introduce a notion of motivic representation stability for stability in the Grothendieck ring of varieties, and show that these character stacks stabilize in this sense for G equal to the general linear groups. Show less
This thesis discusses several questions regarding the double ramification cycle as a Chow class on the moduli space of stable n-pointed genus g curves using tools from so-called logarithmic... Show moreThis thesis discusses several questions regarding the double ramification cycle as a Chow class on the moduli space of stable n-pointed genus g curves using tools from so-called logarithmic geometry. It contains two extracts from articles; the first of these defines the universal double ramification cycle on the Picard stack of n-pointed genus g curves with a line bundle of fixed degree, which is a way to also include the generalisations that are called twisted double ramification cycles. The second article introduces the logarithmic double ramification cycle in the logarithmic Chow ring. The logarithmic double ramification cycle is proven to be ‘logarithmically tautological’ and it helps us prove that the double-double ramification cycle (or the good definition for ‘intersecting double ramification cycles’) is tautological – that is, these classes lie in a subring generated by ‘computable and known’ classes. The second chapter of the thesis explains and illustrates piecewise-polynomial functions, which are key to describing the forementioned ‘logarithmically tautological’, and how these functions relate to classical divisors which we use to describe tautological rings. Show less
In this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are... Show moreIn this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are finite-dimensional, and discuss algorithms that can be used to compute relations among tautological differential forms. Show less
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 thesis consists of three independent chapters. Each chapter deals with a particular link between arithmetic and cohomology. In the first chapter, written together with prof. dr. S.J. Edixhoven... Show moreThis thesis consists of three independent chapters. Each chapter deals with a particular link between arithmetic and cohomology. In the first chapter, written together with prof. dr. S.J. Edixhoven, we consider smooth and proper Deligne-Mumford stacks whose number of points over a finite field is a polynomial. The main result is that the cohomology of such stacks, both etale and Betti, is of Tate type. The second chapter generalizes the p-adic De Rham comparison theorem from schemes to Deligne-Mumford stacks. The last chapter deals with Kedlaya's algorithm for counting points of hyperelliptic curves over finite fields. A different basis than the one described in the original algorithm is described, which has the advantage that it is denominator free. Show less
