Modular forms and K3 surfaces

Modular forms and K3 surfaces

For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over Q associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.

Differential Equations Satisfied by Bi-modular Forms and K3 Surfaces

We study differential equations satisfied by bi-modular forms associated to genus zero subgroups of SL2(R) of the form Γ0(N) or Γ0(N)∗. In some examples, these differential equations are realized as the Picard–Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our method rediscovers some of the Lian–Yau examples of “modular relations” involv...

Differential Equations Satisfied by Modular Forms and K3 Surfaces

We study differential equations satisfied by modular forms of two variables associated to Γ1×Γ2, where Γi (i = 1, 2) are genus zero subgroups of SL2(R) commensurable with SL2(Z), e.g., Γ0(N) or Γ0(N)∗ for some N . In some examples, these differential equations are realized as the Picard–Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our ...

The Enumerative Geometry of K3 Surfaces and Modular Forms

Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating funct...

The Enumerative Geometry of K3 Surfaces and Modular Forms Jim Bryan and Naichung Conan Leung