Two lectures on the arithmetic of K3 surfaces

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update

On the computation of the Picard group for K3K3 surfaces

We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology. This allows to abandon the original limitation to cases of Picard rank $2$ after reduction modulo $p$. Furthermore, the use of Galois data enables us to construct examples which require significantly less computation time

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.Comment: 24 pages, 3 figures, Magma scripts included at the end of the source file

On the arithmetic of a family of degree-two K3 surfaces

Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family $X$.Comment: 20 pages; v2 with some all additions and clarifications suggested by the refere