Strong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational...Show moreStrong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational numbers, a scheme satisfies strong approximation away from S, if the rational points are dense in some product over all p not in S of the sets of p-adic points. In the last century, a couple of great, general results have been proven which give sufficient or necessary conditions for a scheme or a group scheme to satisfy strong approximation away from some set S. In 2019, Kok and Bright showed, using the Brauer-Manin obstruction, that the scheme representing primitive solutions to the equation X1^2 + 47 X2^2 - 103 X3^2 - 103 * 47 * 17 X4^2 = 0 does not satisfy strong approximation away from infinity. On the other hand, in 2020, Pagano and Bright proved a general result from which it follows that this scheme satisfies strong approximation away from infinity, 17, 47 and 103. This thesis shows that this scheme satisfies strong approximation away from infinity and 17, and it obtains a more general result for some equations of the form a X1^2 + b X2^2 + c X3^2 + d X4^2 = 0.Show less
This thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more...Show moreThis thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more intricate is its geometry. Let S be a del Pezzo surface over a field k. It is known that if the degree of S is not 1 and S(k) is non-empty (with a mild extra condition for degree 2), the surface S is k-unirational, meaning that there is a dominant rational map from some projective space to S. If k is an infinite field, this implies that the set S(k) of k-rational points lies Zariski dense in S. But in general, we do not know whether unirationality holds when the degree is 1, and the answer to this question seems way out of reach. So if we want to prove the density of the k-rational points of del Pezzo surfaces of degree 1, we have to search for alternative methods. In this thesis, a result is proven that gives sufficient and necessary conditions for the Zariski density of the rational points on a certain family of del Pezzo surfaces of degree 1.Show less
Adèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals....Show moreAdèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals. In this thesis we define representations of adèles and idèles, enabling us to perform explicit computations in adèle rings and idèle groups. We also discuss two applications of our representations: computing the profinite Fibonacci graph and computing Hilbert class fields of imaginary quadratic number fields using Shimura’s reciprocity law. We implemented these representations as well as the applications in the computer algebra package SageMath.Show less
