On Solvable Automorphism Groups of Function Fields over Finite Fields
Abstract
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 Aut(F/BbbFq), where p=char(BbbFq) and F/BbbFq is a function field of genus gF≥2 with full constant field BbbFq. We prove that there are explicit constants Cq(sol) and Cq(p), depending only on q, such that, for every solvable subgroup G≤Aut(F/BbbFq) and every p-subgroup P≤Aut(F/BbbFq), |G|≤ Cq(sol)(gF−1), |P|≤ Cq(p)(gF−1). In particular, one may take Cq(sol)=12q2(q−1) for q≥4 and Cq(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 Lq>0 and a sequence of function fields Fi/BbbFq with gFi→∞, together with solvable subgroups Gi≤Aut(Fi/BbbFq), such that |Gi|≥ Lq(gFi−1) for all i. For q≥5 one may take Lq=2q(q−1)/(q−2). If q=r2 with r≥3, an unramified tower construction over the Hermitian function field gives the stronger choice Lq=2r3(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 Aut(F/BbbFq) 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.
Related
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BibTeX
@misc{stichtenoth2026solvable,
title = {{On Solvable Automorphism Groups of Function Fields over Finite Fields}},
author = {Henning Stichtenoth and Chaoping Xing and Zugan Xing},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16751},
eprint = {2610.16751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16751},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On Solvable Automorphism Groups of Function Fields over Finite Fields
AU - Henning Stichtenoth
AU - Chaoping Xing
AU - Zugan Xing
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.16751
UR - https://arxiv.org/abs/2610.16751
AB - 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 Aut(F/BbbF_q), where p=char(BbbF_q) and F/BbbF_q is a function field of genus g_F≥2 with full constant field BbbF_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≤Aut(F/BbbF_q) and every p-subgroup P≤Aut(F/BbbF_q), |G|≤ C_q^(sol)(g_F−1), |P|≤ C_q^(p)(g_F−1). In particular, one may take C_q^(sol)=12q^2(q−1) for q≥4 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/BbbF_q with g_F_i→∞, together with solvable subgroups G_i≤Aut(F_i/BbbF_q), such that |G_i|≥ L_q(g_F_i−1) for all i. For q≥5 one may take L_q=2q(q−1)/(q−2). If q=r^2 with r≥3, 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 Aut(F/BbbF_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.
ER -