Automorphisms of Plane Curves defined from Chebychev polynomials

2025/05/02 by Tafazolian, Saeed, Top, Jaap
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.01216

Abstract

In this paper, we study the geometry and automorphism groups of the algebraic curves \(C_d\) defined by the equation \( y^d = φ_d(x) \) over a field \( k \) with \(char(k) \nmid 2d\), where \( φ_d(x) \) is the Chebyshev polynomial of degree \( d \). We classify the total inflection points of \(C_d\), correcting and extending previous work on this. Additionally, we determine the automorphism groups of \(C_d\) 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 \( p=char(k) \). As an application, we use our results to show that certain maximal curves (over finite fields) of the same genus are not isomorphic.

Related

• On certain maximal curves related to Chebyshev polynomials
• On plane curves given by separated polynomials and their automorphisms
• Automorphism group of plane curve computed by Galois points, II
• CM theory, maximal hyperelliptic curves, and Chebyshev polynomials
• Curves defined by Chebyshev polynomials