Automorphisms of Plane Curves defined from Chebychev polynomials

TL;DR

This paper determines the automorphism groups of algebraic curves defined by Chebyshev polynomials over fields with certain characteristics, revealing their symmetries and applications to classifying maximal curves.

Contribution

It explicitly computes automorphism groups for specific degrees and conditions, extending known classifications of these algebraic curves.

Findings

Automorphism groups are fully determined for d=4 and when 2d=p^r+1 or 4d=p^r+1.

For other degrees greater than 4, predictions are made about the automorphism groups.

Certain maximal curves of the same genus are proven not to be isomorphic.

Abstract

We investigate the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = \varphi_d(x), \] where $\varphi_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. We determine the full automorphism group of $\mathcal{C}_d$ in all the cases considered in this paper, namely for $d=4$, and more generally when $2d = p^r+1$ or $4d = p^r+1$. For all other $d>4$, Expectation~\ref{3.19} predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.