On l-class groups of global function fields

Abstract

Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.