Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number (Q1948984)

Let \(D\) be a square-free positive integer and consider \({\mathbb Q} (\sqrt{-D})\). Since Gauss, various authors have studied the ideal class group \(Cl(-D)\) of \({\mathbb Q}(\sqrt{-D})\) and in particular the class number \(h(-D)=|Cl(-D)|\). In particular if \(g\) is a positive integer, let \(N_g(x)\) denote the number of positive square-free \(D \leq x\) such that \(g\mid h(-D)\). \textit{K. Soundararajan} [J. Lond. Math. Soc., II. Ser. 61, No. 3, 681--690 (2000; Zbl 1018.11054)] showed that \[ N_g(x)\gg\begin{cases} x^{\frac{1}{2}+\frac{2}{g}-\varepsilon}&\text{if }g\equiv 0\bmod 4\\ x^{\frac{1}{2}+\frac{3}{g+2}-\varepsilon}&\text{if }g\equiv 2\bmod 4 \end{cases} \] for every \(\varepsilon > 0\). In this paper the authors show a similar result for function fields. The result is: Let \(g\geq 3\) be a fixed integer. Let \(q\) be a power of an odd prime. For odd \(L\), let \(N_g(L)\) denote the number of square-free polynomials \(f\in {\mathbb F}_q[x]\) with \(\deg f \leq L\) such that the class group of a quadratic extension \({\mathbb F}_q(x,\sqrt{f})\) contains an element of order \(g\). Let \(\varepsilon >0\). Then, for all sufficiently large \(L\) we have \[ N_g(L)\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\varepsilon)}. \]