Related papers: On level 2 Modular differential equations

Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation…

Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation…

In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $\mathbf{C}^2$ for integers $m,n\geq2$ and nonzero meromorphic…

Let $P(z)=z^{n}+a_{n-2}z^{n-2}+\cdots+a_0$ be a nonconstant polynomial and $S(z)$ be a nonzero rational function and denote $h(z)=S(z)e^{P(z)}$. Let $\theta\in(0,\pi/2n)$ be a constant and $\varepsilon>0$ be a small constant. It is shown…

Let $f$ be a holomorphic function on the strip $\{z\in C: -\alpha<Im z<\alpha\}, \alpha > 0$, belonging to the class $H(\alpha,-\alpha;\epsilon)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z…

We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In…

A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary…

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we…

In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in…

We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$…

We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $\Gamma_0(4).$

We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method…

Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…

We solve the functional equations $$ \begin{vmatrix} 1 & 1 & 1 f(x) & f(y) & f(z) f\sp{\prime}(x)& f\sp{\prime}(y)& f\sp{\prime}(z) \end{vmatrix} =0,\quad\quad \begin{vmatrix} 1 & 1 & 1 f(x) & g(y) & h(z) \\ f\sp{\prime}(x)&…

Counterexamples to the Modular Isomorphism Problem were discovered recently. These are non-isomorphic finite $2$-groups $G$ and $H$ that have isomorphic group algebras over the field $\mathbb{Z}/2\mathbb{Z}$ and non-isomorphic group…

Certain integrable models are described by pairs (X,Y) of ADET Dynkin diagrams. At high energy these models are expected to have a conformally invariant limit. The S-matrix of the model determines algebraic equations, whose solutions are…

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus…

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations,…

Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy…

Examples of SL(2, Z) actions on differential graded categories are defined and explored.