In this talk, we shall introduce the notion of Shimura’s period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura’s conjecture on the algebraic independence of period symbols. Our results enable us to verify the algebraic independence of the coordinates of any nonzero period vector of an abelian t-module with complex multiplication whose CM type is non-degenerate and defined over an algebraic function field. Further applications will be discussed if time permits. This is joint work with W. Dale Brownawell, Chieh-Yu Chang, and Matthew A. Papanikolas.