Densities for 4-class ranks of quadratic function fields

Abstract

Let F = F q ( 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 K = F ( D ) , where D is a polynomial over F q of odd degree having t distinct irreducible factors. The 4-class rank r 4 ( K ) is the rank of the 4-torsion of the group of divisor classes of K, and it is known that 0 ⩽ r 4 ( K ) ⩽ t − 1 . For fixed r we compute the proportion of such fields K satisfying r 4 ( 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.