Maximal Curves with Few Automorphisms

Abstract

We consider the problem of constructing maximal curves with few automorphisms over finite fields. Our main ingredient is the determination of the \(\operatorname{PGL}_2\!\left(\overline{\mathbb F}_p\right)\)-stabilizer of the roots of the Chebyshev polynomials, apart from two exceptional cases already treated in the literature. We prove that, in all other cases, the only nontrivial M\"obius symmetry is the sign involution. This settles the general case for the automorphism groups of the corresponding Chebyshev curves. We further construct an infinite family of maximal curves of unbounded genus whose only nontrivial automorphism is the hyperelliptic involution. This reveals a contrasting phenomenon to the common heuristic that curves with many rational points tend to have large automorphism groups.