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

TL;DR

This paper explicitly computes the Picard lattice and Galois module structure of a family of degree-two K3 surfaces over the rationals, providing new insights into their arithmetic properties.

Contribution

It offers the first explicit computation of the Picard lattice and Galois structure for this family of K3 surfaces, advancing understanding of their arithmetic and geometric features.

Findings

Explicit Picard lattice computation for the family

Determination of Galois module structure

Enhanced understanding of arithmetic properties

Abstract

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 $X$.