Lower bounds on the number of rational points of Jacobians over finite fields and application to algebraic function fields in towers

We give effective bounds on the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $r\leq g$. Such bounds are especially useful for estimating the class numbers of function fields in towers of function fields over finite fields having several positive Tsfasman–Vlăduţ invariants.