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

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 $\ge 3$, we show that for every $\epsilon >0$, there are $\gg q^{L(1/2+\frac{3}{2(g+1)}-\epsilon)}$ 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 $q^{L(1/2+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue to a similar result of Soundararajan for number fields.