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arXiv · 2505.01216

Automorphisms of Plane Curves defined from Chebychev polynomials

Saeed Tafazolian·Jaap Top
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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.

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Saeed Tafazolian, Jaap Top. 2025-05-02. Automorphisms of Plane Curves defined from Chebychev polynomials. https://arxiv.org/abs/2505.01216

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