K3 surfaces with two involutions and low Picard number

Let X be a complex algebraic K3 surface of degree 2 d and with Picard number ρ ρ . Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ ≥ 1 ρ≥1 when $$d=1$$ d=1 and ρ ≥ 2 ρ≥2 when d ≥ 2 d≥2 . For $$d=1$$ d=1 , the first example defined over Q Q with ρ =1 ρ=1 was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over Q Q , can be used to realise the minimum ρ =2 ρ=2 for all d≥ 2 d≥2 . In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ =2 ρ=2 for $$d=2,3,4$$ d=2,3,4 . We also show that a nodal quartic surface can be used to realise the minimum ρ =2 ρ=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≤ r ≤ 10 1≤r≤10 and signature $$(1,r-1)$$ (1,r-1) there exists a K3 surface Y defined over R R such that \,Pic\,YC=\,Pic\,Y N PicYC=PicY≅N .