Narrow Results

We present a definition of a (super)-modular functor which includes certain interesting cases that previous definitions do not allow. We also introduce a notion of topological twisting of a modular functor, and construct formally a realization by a 2-dimensional topological field theory valued in twisted $K$-modules. We discuss, among other things, the $N=1$-supersymmetric minimal models from the point of view of this formalism.

A simple construction of Eisenstein series for the congruence subgroup Γ₀(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.

We study the continued fractions $I_1(\tau )$ and $I_2(\tau )$ of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction $r(\tau )$ with modularity and many interesting properties. Here we prove the modularities of $I_1(\tau )$ and $I_2(\tau )$ to find the relation with the generator of the field of modular functions on ${\mathrm{\Gamma }}_0(16)$. Moreover we prove that the values $2(I_1{(\tau )}^2+1/I_1{(\tau )}^2)$ and $2(I_2{(\tau )}^2+1/I_2{(\tau )}^2)$ are algebraic integers for certain imaginary quadratic quantity $\tau$.

We study the modularity of the function $u(\tau )=C(\tau )C(2\tau )$, where $C(\tau )$ is Ramanujan’s cubic continued fraction. It is an analogue of Ramanujan’s function $k(\tau )=r(\tau )r{(2\tau )}^2$, where $r(\tau )$ is the Rogers–Ramanujan continued fraction. We first prove the modularity of $u(\tau )$ and express $C(\tau )$ and $C(2\tau )$ in terms of $u(\tau )$. Subsequently, we find modular equations of $u(\tau )$ of level $n$ for every positive integer $n$ by using affine models of modular curves. Finally, we demonstrate that the value of $u(\tau )$ generates the ray class field over an imaginary quadratic field modulo 2 for some $\tau$ in an imaginary quadratic field.

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.