Designs, Codes and Cryptography

, Volume 39, Issue 1, pp 33–37 | Cite as

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms



The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve \(\mathcal{X}\) then either \(\deg(\mathcal{X})=q+2\) or \(\deg(\mathcal {X})\ge q^2+q+1\). In the former case \(\mathcal{X}\) is projectively equivalent to the curve \(\mathcal{X}_q\) with equation Xq+1Y+Yq+1+X=0 studied by Pellikaan. Furthermore, the curve \(\mathcal{X}_q\) has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its \(\mathbb{F}_{{q}^{3}}\)-rational points can be partitioned into finite projective planes \(P^{2}(\mathbb{F}_{q})\). In this paper, the full automorphism group of such curves is determined. It turns out that \(Aut(\mathcal {X}_q)\) is the normalizer of a Singer group in \(PGL(3,\mathbb{F}_{q})\).


algebraic curve singer cyclic group automorphisms 

AMS Classification

14H37 14H50 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità della BasilicataPotenzaItaly

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