Densities for 4-class ranks of quadratic function fields

Let F=Fq(T) be a rational function field of odd characteristic, and fix a positive integer t. In this article we study the family of quadratic function fields , where D is a polynomial over Fq of odd degree having t distinct irreducible factors. The 4-class rank r4(K) is the rank of the 4-torsion of the group of divisor classes of K, and it is known that 0⩽r4(K)⩽t−1. For fixed r we compute the proportion of such fields K satisfying r4(K)=r, and in particular we determine the behaviour of this value as t→∞. We will need some asymptotic results for these computations, in particular the number of polynomials D as above whose irreducible factors fulfill certain parity and quadratic residue conditions.