Lower bounds on the number of rational points of Jacobians over finite fields and application to algebraic function fields in towers (Q2801886)

The authors prove effective lower bounds on the class number of any algebraic function field of genus \(g\) over a finite field. The bound requires the number of degree \(r\) places for some values of \(r \leq g\). The authors claim that their method offers an improvement over previous work if these counts are known for one degree \(r > 1\), several degrees \(\geq 1\), or only for \(r=1\) and sufficiently large \(g\). As a demonstration of the method, lower bounds are computed on the class numbers of the first two function fields in the tower of \textit{F. Hess} et al. [J. Algebra Appl. 12, No. 4, Paper No. 1250190, 23 p. (2013; Zbl 1310.11111)].