Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

Abstract

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field \(\mathbb{F}_{q}(x)\) whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥ 3, we show that for every ε > 0, there are \(\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}\) polynomials \(f \in \mathbb{F}_{q}[x]\) with \(\deg f=L\), for which the class group of the quadratic extension \(\mathbb{F}_{q}(x, \sqrt{f})\) has an element of order g. This sharpens the previous lower bound \(q^{L(\frac{1}{2}+\frac{1}{g})}\) of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.