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We give lower bounds on the number of effective divisors of degree $\leq g-1$with respect to the number of places of certain degrees of an algebraicfunction field of genus $g$ defined over a finite field. We deduce lower boundsand asymptotics for the class number, depending mainly on the number of placesof a certain degree. We give examples of towers of algebraic function fieldshaving a large class number.

Lower bounds on the class number of algebraic function fields defined over any finite field