Nonlinear differential equations satisfied by certain classical modular forms

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 ...

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...

ON TE EXISTENCE OF PERIODIC SOLUTION FOR CERTAIN NONLINEAR THIRD ORDER DIFFERENTIAL EQUATIONS

Local and global Maass relations

We characterize the irreducible, admissible, spherical representations of GSp4(F ) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-...

Local and global Maass relations (expanded version)

We characterize the irreducible, admissible, spherical representations of GSp4(F ) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-...