This talk will concern the divisibility of class numbers and their relation with Bernoulli numbers for function fields of positive characteristic. In number fields, the Herbrand-Ribet theorem gives a precise relation between the divisibility between class numbers and Bernoulli numbers. In function fields, the Herbrand's direction is proved to be true, but the other direction has obvious conuter-examples. Gekeler reformulated a conjecture similar to Ribet's theorem. This talk will report some recent progress towards the Gekeler's conjecture.