On Solvable Automorphism Groups of Function Fields over Finite Fields

The classical Hurwitz bound gives a linear upper bound for automorphism groups of algebraic curves in characteristic zero, whereas algebraic curves in positive characteristic can have much larger automorphism groups. In this article, we firstly investigate the solvable subgroups and $p$-subgroups of $\mathrm{Aut}(F\mathrm{/}{\mathbb{F}}_q)$ where $p=\mathrm{char}({\mathbb{F}}_q)$ and $F\mathrm{/}{\mathbb{F}}_q$ is a function field of genus $g_F\ge 2$ with full constant field ${\mathbb{F}}_q$. We prove that there are explicit constants $C_q^{(\mathrm{s}\mathrm{o}\mathrm{l})}$ and $C_q^{(p)}$, depending only on $q$, such that, for every solvable subgroup $G\le \mathrm{Aut}(F\mathrm{/}{\mathbb{F}}_q)$ and every $p$-subgroup $P\le \mathrm{Aut}(F\mathrm{/}{\mathbb{F}}_q)$, $$\mathrm{\mid }G\mathrm{\mid }\le C_q^{(\mathrm{s}\mathrm{o}\mathrm{l})}(g_F-1),\mathrm{\mid }P\mathrm{\mid }\le C_q^{(p)}(g_F-1)\mathrm{.}$$ In particular, one may take $C_q^{(\mathrm{s}\mathrm{o}\mathrm{l})}=12q^2(q-1)$ for $q\ge 4$ and $C_q^{(p)}=5q$ for every $q$. In the second part of this article , we give the asymptotic lower bounds for solvable automorphism groups. For every $q$ there is an explicit constant $L_q>0$ and a sequence of function fields $F_i\mathrm{/}{\mathbb{F}}_q$ with $g_{F_i}\to \mathrm{\infty }$, together with solvable subgroups $G_i\le \mathrm{Aut}(F_i\mathrm{/}{\mathbb{F}}_q)$, such that $$\mathrm{\mid }{G}_i\mathrm{\mid }\ge L_q(g_{F_i}-1)\mathrm{.}$$ For $q\ge 5$ one may take $L_q=2q(q-1)\mathrm{/}(q-2)$. If $q=r^2$ with $r\ge 3$, an unramified tower construction over the Hermitian curve gives the stronger choice $L_q=2r^3(r-1)\mathrm{/}(r-2)$. Moreover, as an application of our tower construction above, we improve the lower bound of Ma and Xing to $B_q\ge 2(q-1)\mathrm{/}(q-2)$ for $q\ge 5$. As another application, we extend our method for bounding solvable automorphism groups to $p$-solvable subgroups, obtaining explicit linear bounds in terms of the genus.