The $L$-polynomial of hyperelliptic function fields and its applications

Let $\mathrm{\ell }$ be an odd prime, $q$ an odd prime power such that $q\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell })$, and $m$ the order of $q$ in ${\text{F}}_{\mathrm{\ell }}^{\times }$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:={\text{F}}_q(T,\sqrt[\mathrm{\ell }]{T^2+aT+b})$ with $a,b\in {\text{F}}_q$ and $a^2-4b\mathrm{\ne }0$. Using our formula, we obtain the explicit closed formula for the class number of $K$, where $m$ is even or $m=\frac{\mathrm{\ell }-1}{2}$.As an application, we compute the average class numbers for hyperelliptic function fields with genus up to $3$.