有限域上函数域的可解自同构群
On Solvable Automorphism Groups of Function Fields over Finite Fields
AI总结：
本文研究有限域上亏格≥2的函数域的可解自同构群，给出其可解子群与p-子群的显式线性上界、可解自同构群的渐近下界，并将方法推广到p-可解子群，还改进了Ma与Xing的相关下界。
摘要：
经典Hurwitz界给出了特征零下代数曲线自同构群的线性上界，而特征为正的代数曲线可拥有大得多的自同构群。本文中，我们首先研究Aut(F/𝔽_q)的可解子群与p-子群，其中p=char(𝔽_q)，F/𝔽_q是亏格g_F≥2且具有全常数域𝔽_q的函数域。我们证明存在仅依赖q的显式常数C_q^(sol)与C_q^(p)，使得对每个可解子群G≤Aut(F/𝔽_q)和每个p-子群P≤Aut(F/𝔽_q)，有|G|≤C_q^(sol)(g_F-1)，|P|≤C_q^(p)(g_F-1)。特别地，当q≥4时可取C_q^(sol)=12q²(q-1)，对任意q可取C_q^(p)=5q。本文第二部分，我们给出可解自同构群的渐近下界：对每个q，存在显式常数L_q>0及一列函数域F_i/𝔽_q，满足g_F_i→∞，同时存在可解子群G_i≤Aut(F_i/𝔽_q)，使得|G_i|≥L_q(g_F_i-1)。当q≥5时可取L_q=2q(q-1)/(q-2)；若q=r²且r≥3，通过Hermitian曲线上的非分歧塔构造可得到更优的L_q=2r³(r-1)/(r-2)。此外，作为上述塔构造的应用，我们将Ma与Xing的下界改进为B_q≥2(q-1)/(q-2)（q≥5）；作为另一应用，我们将可解自同构群的界方法推广到p-可解子群，得到了关于亏格的显式线性界。
英文摘要：
The classical Hurwitz bound gives a linear upper bound in terms of the genus for automorphism groups of algebraic function fields in characteristic zero, whereas algebraic function fields in positive characteristic can have much larger automorphism groups. Usually, one considers the case where the constant field is an algebraically closed field. Here we focus on function fields over a finite constant field. It turns out that the situation is very different. We firstly investigate solvable subgroups and $p$-subgroups of $\rm{Aut}(F/\Bbb{F}_q)$, where $p=\rm{char}(\Bbb{F}_q)$ and $F/\Bbb{F}_q$ is a function field of genus $g_F\ge2$ with full constant field $\Bbb{F}_q$. We prove that there are explicit constants $C_q^{(sol)}$ and $C_q^{(p)}$, depending only on $q$, such that, for every solvable subgroup $G\le\rm{Aut}(F/\Bbb{F}_q)$ and every $p$-subgroup $P\le\rm{Aut}(F/\Bbb{F}_q)$, $$ |G|\le C_q^{(sol)}(g_F-1),\qquad |P|\le C_q^{(p)}(g_F-1).$$ In particular, one may take $C_q^{(sol)}=12q^2(q-1)$ for $q\ge4$ and $C_q^{(p)}=5q$ for every $q$. In the second part of this article, we give asymptotic lower bounds for solvable automorphism groups. For every $q$ there is an explicit constant $L_q>0$ and a sequence of function fields $F_i/\Bbb{F}_q$ with $g_{F_i}\to\infty$, together with solvable subgroups $G_i\le\rm{Aut}(F_i/\Bbb{F}_q)$, such that $$ |G_i|\ge L_q(g_{F_i}-1) $$ for all $i$. For $q\ge5$ one may take $L_q=2q(q-1)/(q-2)$. If $q=r^2$ with $r\ge3$, an unramified tower construction over the Hermitian function field gives the stronger choice $L_q=2r^3(r-1)/(r-2)$. Moreover, as an application of our tower construction above, we improve the asymptotic lower bound of Ma and Xing for subgroups of $\rm{Aut}(F/\Bbb{F}_q)$ whose order is relatively prime to $p$. As another application, we extend our method to obtain explicit linear bounds for $p$-solvable groups in terms of the genus.
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