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An explicit eight-dimensional family of K3 surfaces with real multiplication by Q(√5)

Year: 2026
Type: preprint
Abstract: For a K3 surface whose transcendental lattice has real multiplication (RM), the Hodge conjecture for its self-product comes down to the algebraicity of that multiplication. Families of K3 surfaces with RM by a real quadratic field have at most eight moduli; such families exist by lattice theory, but the existence results give no equations, and van Geemen and Schütt, who constructed a seven-dimensional family with RM by ℚ(√5), noted that they did not know the general member of an eight-dimensional one explicitly. We write down such a family: double planes w² = F, where the plane sextic F is built from a binary sextic, a binary cubic and one further parameter subject to a quadratic relation. It contains the family of van Geemen and Schütt; its very general member S is a K3 surface of degree two with Picard number 2 and endomorphism field ℚ(√5), and its image is dense in an eight-dimensional irreducible component of the locus of K3 surfaces of degree two with RM by ℚ(√5). The lines through one point of the plane cut S in a pencil of genus-two curves that deform the Dickson-type curves of Tautz, Top and Verberkmoes. Using the Elkies–Kumar equations for the Humbert surface and the Hilbert modular surface of discriminant 5, we show that their Jacobians have RM by the ring of integers of ℚ(√5), defined over the base of the pencil, and a general argument shows that such a pencil on any K3 surface induces RM on its transcendental lattice by an algebraic cycle. Consequently the Hodge conjecture holds for S × S and for the Hilbert square of S. To our knowledge this is the first explicit family of maximal dimension with RM by a real quadratic field. We do not treat the exceptional Hodge classes needed for the higher powers of S. The family was found by a computer search over a finite field; the proofs use exact computer checks (identities between Igusa–Clebsch invariants, made with two independent implementations; smoothness; the number of moduli). Code and data are at github.com/sallerk/HC-k3-rm5-maximal.
DOI: 10.5281/zenodo.23309348
Language: English
Open Access status: green
OpenAlex ID: W7221509223
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