For F/K an algebraic function field in one variable over a finite field of constants K (i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K), let N(F ) and g(F ) denote the number of places of degree one and the genus, respectively, of F . Let F = (F1, F2, F3, . . .) be a tower of function fields, each defined over K. Further, we will assume that F1 ⊆ F2 ⊆ F3 . ...