K3 surfaces with two involutions and low Picard number

Let X be a complex algebraic K3 surface of degree 2d and with Picard number \(\rho \). Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, \(\rho \ge 1\) when \(d=1\) and \(\rho \ge 2\) when \(d \ge 2\). For \(d=1\), the first example defined over \({\mathbb {Q}}\) with \(\rho =1\) was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over \({\mathbb {Q}}\), can be used to realise the minimum \(\rho =2\) for all \(d\ge 2\). In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum \(\rho =2\) for \(d=2,3,4\). We also show that a nodal quartic surface can be used to realise the minimum \(\rho =2\) for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank \(1\le r \le 10\) and signature \((1,r-1)\) there exists a K3 surface Y defined over \({\mathbb {R}}\) such that \({{\,\textrm{Pic}\,}}Y_{\mathbb {C}}={{\,\textrm{Pic}\,}}Y \cong N\).