An explicit five-dimensional family of K3 surfaces with real multiplication by Q(√2), and the limits of the genus-two pencil method

In an earlier note, real multiplication by Q(√5), induced by an algebraic cycle, was obtained on an eight-dimensional family of K3 double planes from a pencil of genus-two curves whose Jacobians have real multiplication over the base of the pencil. This note asks which other real quadratic fields the mechanism reaches. For Q(√2) it gives an explicit five-dimensional family of double planes branched along a line and a quintic: the lines through a point of the quintic cut out quadratic twists of curves of Tautz, Top and Verberkmoes, whose Jacobians have real multiplication by Z[√2] over the base. On every member √2 acts on the transcendental lattice through an algebraic cycle, and exact Frobenius traces show that the very general member has Picard number six. With two earlier, unrefereed notes, this gives the Hodge conjecture for the powers S^n, n ≤ 7, and the Hilbert schemes S^[n], n ≤ 35. In the sources read, every earlier explicit family with real multiplication by Q(√2) has dimension at most four and Picard number at least ten. The note then bounds the method. Real multiplication over the base forces the local monodromy at each singular fibre to commute with it, which excludes some degenerations and fixes the order of the discriminant at the others. On the equations of Elkies and Kumar for Hilbert modular surfaces this shows that, for a smooth branch curve, a centre off it and the maximal order, the method works among the five discriminants below 100 with a rational Hilbert modular surface only for Q(√5). For Z[√2], branch curves with simple singularities and a centre that is not a singular point of the curve, it gives at most six moduli, while the period domain has dimension eight; one configuration with six moduli remains open. Code and data are at github.com/sallerk/HC-k3-rm-fields.