Automorphisms of Plane Curves defined from Chebychev polynomials
Saeed Tafazolian, Jaap Top
TL;DR
This work analyzes plane curves ${\mathcal{C}}_d$ defined by $y^d=\varphi_d(x)$ where $\varphi_d$ is the Chebyshev polynomial and $\operatorname{char}(k)\nmid 2d$. It first determines the total inflection points, showing they are the $d$ points with $y=0$ unless the exceptional case $\operatorname{char}(k)=p\ge3$ and $d=(p^m+1)/2$ yields $3d$ points. It then classifies $\operatorname{Aut}({\mathcal{C}}_d)$ in key regimes: in the Fermat-like case $2d=q+1$ ($q=p^m$) the automorphism group is the full Fermat-type group $(\mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}/d\mathbb{Z})\rtimes S_3$, while in the generic case the group fits into a short exact sequence with $N\cong \mathbb{Z}/d\mathbb{Z}$ and a quotient acting via automorphisms of $\mathbb{P}^1$ permuting the zeros of $\varphi_d$. The authors provide explicit structures in special cases (notably $d=4$ and $4d=q+1$) and apply these results to construct families of maximal curves over $\mathbb{F}_{q^2}$ of the same genus that are not isomorphic, refining previous work (ABS) and illustrating how automorphism data distinguish maximal curves.
Abstract
In this paper, we study the geometry and automorphism groups of the algebraic curves \(\mathcal{C}_d\) defined by the equation \( y^d = \varphi_d(x) \) over a field \( k \) with \(\operatorname{char}(k) \nmid 2d\), where \( \varphi_d(x) \) is the Chebyshev polynomial of degree \( d \). We classify the total inflection points of \(\mathcal{C}_d\), correcting and extending previous work on this. Additionally, we determine the automorphism groups of \(\mathcal{C}_d\) in several cases, namely for \( d=4 \), and for any $d$ such that \( 2d = q+1 \) or \( 4d = q+1 \) for an arbitrary power \( q \) of the prime \( p=\operatorname{char}(k) \). As an application, we use our results to show that certain maximal curves (over finite fields) of the same genus are not isomorphic.