Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface

Abstract:

A $K3$ surface $X$ defined over a field $k$ of characteristic $0$ is called singular if the Néron-Severi lattice $\mathrm {NS}(X)$ of $X\otimes \overline {k}$ is of rank $20$. Let $X$ be a singular $K3$ surface defined over a number field $F$. For each embedding $\sigma : F\hookrightarrow \mathbb {C}$, we denote by $T(X^\sigma )$ the transcendental lattice of the complex $K3$ surface $X^\sigma$ obtained from $X$ by $\sigma$. For each prime $\mathfrak {p}$ of $F$ at which $X$ has a supersingular reduction $X_{\mathfrak {p}}$, we define $L(X, \mathfrak {p})$ to be the orthogonal complement of $\mathrm {NS}(X)$ in $\mathrm {NS}(X_{\mathfrak {p}})$. We investigate the relation between these lattices $T(X^ \sigma )$ and $L(X,\mathfrak {p})$. As an application, we give a lower bound for the degree of a number field over which a singular $K3$ surface with a given transcendental lattice can be defined.