  In this paper we 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.
This is an extension of Yu's work on Hilbert-Blumenthal t-modules.
