Vector-valued modular functions for the modular group and the hypergeometric equation

Vector valued hermitian and quaternionic modular forms

A “strange” Vector-valued Quantum Modular Form

Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of Q, and whose obstruction to modularity is some a...

The matrix-valued hypergeometric equation.

The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11-13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123-162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39-83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127-172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1-24]. The hypergeometric function kno...

Ruijsenaars ’ Hypergeometric Function and the Modular Double

Simultaneous eigenfunctions of two Askey-Wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra Uq(sl 2 (C)). These eigenfunctions are shown to be equal to Ruijsenaars' hypergeometric function under a proper parameter correspondence .

Structure of the module of vector-valued modular forms

Let V be a representation of the modular group Γ of dimension p. We show that the Z-graded space H(V ) of holomorphic vector-valued modular forms associated to V is a free module of rank p over the algebra M of classical holomorphic modular forms. We study the nature of H considered as a functor from Γ-modules to graded M-lattices and give some applications, including the calculation of the Hil...