On nilpotent automorphism groups of function fields

We study the automorphisms of a function field of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. More precisely, we show that the order of a nilpotent subgroup $G$ of its automorphism group is bounded by $16 (g-1)$ when G is not a $p$-group. We show that if $|G|=16(g-1)$, then $g-1$ is a power of $2$. Furthermore, we provide an infinite family of function fields attaining the bound.