Remarks on $q$-exponents of generalized modular functions

Abstract

We prove several multiplicity one theorems for generalized modular functions (GMF), in terms of their $q$-exponents, and make a precise statement about the nature of values that the prime $q$-exponents of a GMF can take. We shall also study the integrality of general $q$-exponents of a GMF and give an upper bound on the first sign change of these $q$-exponents.