l-Class groups of cyclic function fields of degree l

Let l be a prime number and K be a cyclic extension of degree l of the rational function field Fq(T) over a finite field of characteristic ≠l. We study the l-part of the ideal class group of the integral closure of Fq[T] in K, and the l-part of the group of divisor classes of degree 0 of K as Galois modules. Using class field theory, we can describe explicitly part of the structure of these l-class groups. As an application, we get (for l=2) bounds for the order of the 4-torsion on JX(Fq), the group of points defined over Fq on the Jacobian of a hyperelliptic curve X/Fq.