L-Class groups of cyclic function fields of degree l

Abstract

Let l be a prime number and K be a cyclic extension of degree l of the rational function field F q ( T ) over a finite field of characteristic ≠ l. We study the l-part of the ideal class group of the integral closure of F q [ 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 J X ( F q ) , the group of points defined over F q on the Jacobian of a hyperelliptic curve X / F q .