Uniqueness of the maximal function field with the third largest genus in characteristic $3$

Source abstract

Let $n$ be a positive integer and define $q=3^n$. It is well known that there exists a maximal function field $Z_3$ over $\mathbb{F}_{q^2}$ of genus $g_3=q(q-1)/6$ and that $g_3$ is the third largest genus a maximal function field over $\mathbb{F}_{q^2}$ can have. In this article we show that any maximal function field over $\mathbb{F}_{q^2}$ of genus $g_3$ is isomorphic to $Z_3$. This extends previously known uniqueness result for the largest and second largest genera of maximal curves.