Proof of the inverse period map conjecture for polarized K3 surfaces

Prove that the explicitly constructed orthogonal-modular-form coefficient functions for the elliptic K3 family define an explicit inverse period map for algebraic K3 surfaces polarized by the lattice U\oplus E_8(-1).

Background

The paper constructs an elliptic K3 family whose coefficients are orthogonal modular forms for the even unimodular lattice of signature (2,10). The construction is fixed by imposing a scaling limit that reproduces the Seiberg–Witten curve of the E-string theory, and it passes consistency checks at special Wilson-line values, including the known inverse period map for (U\oplus E_8(-1)\oplus E_7(-1))-polarized K3 surfaces.

Despite these checks, the authors only conjecture that the resulting coefficient functions give the inverse period map for (U\oplus E_8(-1))-polarized K3 surfaces. They explicitly leave a proof unresolved and suggest that computing the periods of the constructed K3 surfaces would be a natural approach, although this may require further technical advances.

References

We therefore conjecture that our result gives an explicit inverse period map for $(U\oplus E_8(-1))$-polarized K3 surfaces. We leave the proof as an open problem for future work.

— F-theory on K3 surfaces and orthogonal modular forms  (2609.10669 - Sakai, 9 Sep 2026) in Section 5, “Construction of the solution”; Section 7, “Conclusions and outlook”