On generalizations of Fermat curves over finite fields and their automorphisms

ABSTRACT

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽_q of characteristic p>2. Assume that the 𝔽_q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups C_m and C_n of orders m and n prime to p, such that both quotient curves 𝒳∕C_n and 𝒳∕C_m are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups.

KEYWORDS:

• Algebraic curves
• automorphism groups
• finite fields

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11G20, 14H05, 14H37
• Secondary: 14HXX

Acknowledgment

The authors would like to thank Gábor Korchmáros and Massimo Giulietti for many useful conversations on the topic of this article.

Notes

¹The only situation in which a cyclic subgroup of PGL(2,q) of order n|q+1 fixes an 𝔽_q-rational point is n = 2.