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
