On the arithmetic of a family of degree – two \(K3\) surfaces. (English) Zbl 1410.14029

The paper under review proves the following theorem:
Theorem. Let \({\mathbb{P}}={\mathbb{P}}_{\mathbb{Q}}(1,1,1,3)\) denote the weighted projective space over \({\mathbb{Q}}\) with weights \((1,1,1,3)\) and coordinates \(x,y,z\) and \(w\). Let \({\mathbb{A}}^1\) be the affine line over \({\mathbb{Q}}\), with coordinate \(t\). Consider the family of degree-two \(K3\) surfaces \(X: w^2=x^6+y^6+x^6+tx^2y^2z^2\). Let \({\mathfrak{X}}\) be the generic element of this family, and let \(\overline{\mathfrak{X}}\) be its base change to \(\overline{\mathbb{Q}(t)}\).
Then \(\mathfrak{X}\) is a \(K3\) surface defined over \({\mathbb{Q}}(t)\). The geometric Picard lattice \(\text{Pic}\overline{\mathfrak{X}}\) is isomorphic to the unique (up to isometries) lattice with rank \(19\), signature \((1,18)\), determinant \(2^53^3\), and discriminant group isomorphic to \({\mathbb{Z}}/6{\mathbb{Z}}\times (\mathbb{Z}/12{\mathbb{Z}})^2\).
The theorem is proved, first by showing that \({\mathfrak{X}}\) is isogenous to the Kummer surface associated to the abelian surface \(E\times E\) where \(E\) is an elliptic curve with \(j\)-invariant \(-(4t)^3\). Then the geometric Picard number \(\rho(\overline{\mathfrak{X}})\) is computed to be \(19\) and the Picard lattice is explicitly determined. The explicit description of \(\text{Pic}\overline{\mathfrak{X}}\) enabled them to find the arithmetic invariants.


14J28 \(K3\) surfaces and Enriques surfaces


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