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 PhD thesis concerns the topic of arithmetic geometry. We address three different questions and each of the questions in some way is about counting how big some set is or can be. We produce... Show moreThis PhD thesis concerns the topic of arithmetic geometry. We address three different questions and each of the questions in some way is about counting how big some set is or can be. We produce heuristics for counting rational points on surfaces given by one diagonal quartic equation. Our results match with experimental data obtained by van Luijk a few years ago. A different result concerns a certain type of conic bundles over low degree hypersurfaces. We count rational points on the base over which the fibre has rational points. We are able to produce asymptotic results where most results in the literature only produce upper bounds. Moreover we investigate the leading constant in this asymptotic formula, matching it up with expected conjectural behaviour that can be found in the literature. Lastly, we study Brauer groups of Kummer surfaces. We give a way to obtain upper bounds for their sizes. Our way is effective (one only needs to use a formula), but the bounds obtained seem not to be sharp. Our method is based on effective versions of Faltings' theorem on finiteness of abelian varieties. Show less
In this thesis we study the unirationality of del Pezzo surfaces of degree 2 over finite fields, proving that every such surface is unirational. We explicitly compute the Picard lattice of... Show more In this thesis we study the unirationality of del Pezzo surfaces of degree 2 over finite fields, proving that every such surface is unirational. We explicitly compute the Picard lattice of the members of a 1-dimensional family of K3 surfaces. We produce an explicit example of a K3 surface having a particular Picard lattice of rank 2. Show less
A common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence... Show moreA common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence results concerning K3 surfaces defined over the rational numbers with a dense set of rational points with respect to the p-adic topology, for a prime number p, and product topologies arising from these Show less
