Modular forms and K3 surfaces

The Langlands Program and String Modular K3 Surfaces

A number theoretic approach to string compactification is developed for CalabiYau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic two-form of a particular type of K3 surfaces can be expressed in terms of modular forms constructed from the worldsheet theory. The process of...

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

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 ...

K3 K3 K3 Surfaces with Involution and Analytic Torsion

In a series of works [Bo3-5], Borcherds developed a theory of modular forms over domains of type IV which admits an infinite product expansion. Such modular forms are said to be Borcherds's product in this paper. Among all Borcherds's products, Borcherds's Φ-function ([Bo4]) has an interesting geometric background; It is a modular form on the moduli space of Enriques surfaces characterizing the...