The asymptotic behavior of automorphism groups of function fields over finite fields
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- by Liming Ma and Chaoping Xing PDF
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Abstract:
The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity.
Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group $\textrm {Aut}(F/{\mathbb F}_q)$ in terms of genus ${g_F}$ for a function field $F$ over a finite field ${\mathbb F}_q$. Although the whole group $\textrm {Aut}(F/{\mathbb F}_q)$ could have size $\Omega ({g_F}^4)$, the maximum size $m_F$ of abelian subgroups of the automorphism group $\textrm {Aut}(F/{\mathbb F}_q)$ is upper bounded by $4g_F+4$ for $g_F\ge 2$. In the present paper, we study the asymptotic behavior of $m_F$ by defining $M_q=\limsup _{{g_F}\rightarrow \infty }\frac {m_F \cdot \log _q m_F}{{g_F}}$, where $F$ runs through all function fields over ${\mathbb F}_q$. We show that $M_q$ lies between $2$ and $3$ (resp., $4$) for odd characteristic (resp., even characteristic). This means that $m_F$ grows much more slowly than genus does asymptotically.
The second part of this paper is to study the maximum size $b_F$ of subgroups of $\textrm {Aut}(F/{\mathbb F}_q)$ whose order is coprime to $q$. The Hurwitz bound gives an upper bound $b_F\le 84(g_F-1)$ for every function field $F/{\mathbb F}_q$ of genus $g_F\ge 2$. We investigate the asymptotic behavior of $b_F$ by defining ${B_q}=\limsup _{{g_F}\rightarrow \infty }\frac {b_F}{{g_F}}$, where $F$ runs through all function fields over ${\mathbb F}_q$. Although the Hurwitz bound shows ${B_q}\le 84$, there are no lower bounds on $B_q$ in the literature. One does not even know whether ${B_q}=0$. For the first time, we show that ${B_q}\ge 2/3$ by explicitly constructing some towers of function fields in this paper.
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Additional Information
- Liming Ma
- Affiliation: School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China
- MR Author ID: 1034471
- Email: lmma@yzu.edu.cn
- Chaoping Xing
- Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
- MR Author ID: 264368
- Email: xingcp@ntu.edu.sg
- Received by editor(s): January 9, 2018
- Published electronically: March 26, 2019
- Additional Notes: The first author was partially supported by the National Natural Science Foundation of China under Grant 11501493, the Shuangchuang Doctor Project of Jiangsu Province, and the Young Visiting Scholar Project of the China Scholarship Council.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 35-52
- MSC (2010): Primary 14H37, 11R58
- DOI: https://doi.org/10.1090/tran/7822
- MathSciNet review: 3968761