On maximal plane curves of degree $3$ over $\mathbb{F}_4$, and Sziklai's example of degree $q-1$ over $\mathbb{F}_q$
Keywords:
Plane curve, Finite field, Rational point, Maximal curve
Abstract
An elementary and self-contained argument for the complete determination of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal plane curves of degree $3$ over $\mathbb{F}_4$ up to projective equivalence over $\mathbb{F}_4$ appear, they are birationally equivalent over $\mathbb{F}_4$ each other.
References
\begin{thebibliography}{20}
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{aub-per1996}
\href{https://www.math.univ-toulouse.fr/~perret/Fichiers/Scan-Weil.Singulier.pdf}{
Y. Aubry and M. Perret,
A Weil theorem for singular curves,
in: R. Pellikaan, M. Perret and S. Vl\u{a}du\c{t} (Eds.),
Arithmetic geometry and coding theory (Luminy, 1993),
de Gruyter, Berlin, 1996, 1--7. }
%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hir}
J. W. P. Hirschfeld,
Projective geometries over finite fields
(second edition),
Oxford University Press,
Oxford, 1998.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hir-sto-tha-vol1991}
\href{https://link.springer.com/article/10.1007/BF01258509}
{J. W. P. Hirschfeld, L. Storme, J. A. Thas and J. F. A. Voloch,
A characterization of Hermitian curves,
J. Geom. 41 (1991) 72--78.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2009}
\href{https://www.sciencedirect.com/science/article/pii/S1071579709000215}
{M. Homma and S. J. Kim,
Around Sziklai's conjecture on the number of points of
a plane curve over a finite field,
Finite Fields Appl. 15 (2009), 468-474.}
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2010a}
\href{https://arxiv.org/pdf/0907.1325.pdf}
{M. Homma and S. J. Kim,
Sziklai's conjecture on the number of points of
a plane curve over a finite field {\rm II},
in: G. McGuire, G.L. Mullen, D. Panario, I.E. Shparlinski (Eds.),
Finite Fields: Theory and Applications, 225--234,
Contemp. Math., vol. 518, AMS,
Providence, 2010.
(An update is available at arXiv 0907.1325v2.)}
%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2010b}
\href{https://www.sciencedirect.com/science/article/pii/S107157971000050X}
{M. Homma and S. J. Kim,
Sziklai's conjecture on the number of points of
a plane curve over a finite field {\rm III},
Finite Fields Appl. 16 (2010) 315--319.}
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{ruc-sti1994}
\href{https://doi.org/10.1515/crll.1994.457.185}
{H.-G. R\"{u}ck and H. Stichtenoth,
A characterization of Hermitian function fields over finite fields,
J. Reine Angew. Math. 457 (1994) 185--188. }
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{sch}
\href{https://doi.org/10.1016/0097-3165(87)90003-3}
{R. Schoof,
Nonsingular plane cubic curves over finite fields,
J. Combin. Theory Ser. A 46 (1987) 183--211.}
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{ste2012}
B. Steinberg,
Representation theory of finite groups.
An introductory approach, Universitext, Springer, New York, 2012.
%%%%%%%%%%%%%%%%%%%%%
\bibitem{szi2008}
\href{https://doi.org/10.1016/j.ffa.2007.09.004}
{P. Sziklai,
A bound on the number of points of a plane curve,
Finite Fields Appl. 14 (2008) 41--43.}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{aub-per1996}
\href{https://www.math.univ-toulouse.fr/~perret/Fichiers/Scan-Weil.Singulier.pdf}{
Y. Aubry and M. Perret,
A Weil theorem for singular curves,
in: R. Pellikaan, M. Perret and S. Vl\u{a}du\c{t} (Eds.),
Arithmetic geometry and coding theory (Luminy, 1993),
de Gruyter, Berlin, 1996, 1--7. }
%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hir}
J. W. P. Hirschfeld,
Projective geometries over finite fields
(second edition),
Oxford University Press,
Oxford, 1998.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hir-sto-tha-vol1991}
\href{https://link.springer.com/article/10.1007/BF01258509}
{J. W. P. Hirschfeld, L. Storme, J. A. Thas and J. F. A. Voloch,
A characterization of Hermitian curves,
J. Geom. 41 (1991) 72--78.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2009}
\href{https://www.sciencedirect.com/science/article/pii/S1071579709000215}
{M. Homma and S. J. Kim,
Around Sziklai's conjecture on the number of points of
a plane curve over a finite field,
Finite Fields Appl. 15 (2009), 468-474.}
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2010a}
\href{https://arxiv.org/pdf/0907.1325.pdf}
{M. Homma and S. J. Kim,
Sziklai's conjecture on the number of points of
a plane curve over a finite field {\rm II},
in: G. McGuire, G.L. Mullen, D. Panario, I.E. Shparlinski (Eds.),
Finite Fields: Theory and Applications, 225--234,
Contemp. Math., vol. 518, AMS,
Providence, 2010.
(An update is available at arXiv 0907.1325v2.)}
%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hom-kim2010b}
\href{https://www.sciencedirect.com/science/article/pii/S107157971000050X}
{M. Homma and S. J. Kim,
Sziklai's conjecture on the number of points of
a plane curve over a finite field {\rm III},
Finite Fields Appl. 16 (2010) 315--319.}
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{ruc-sti1994}
\href{https://doi.org/10.1515/crll.1994.457.185}
{H.-G. R\"{u}ck and H. Stichtenoth,
A characterization of Hermitian function fields over finite fields,
J. Reine Angew. Math. 457 (1994) 185--188. }
%%%%%%%%%%%%%%%%%%%%%%
\bibitem{sch}
\href{https://doi.org/10.1016/0097-3165(87)90003-3}
{R. Schoof,
Nonsingular plane cubic curves over finite fields,
J. Combin. Theory Ser. A 46 (1987) 183--211.}
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{ste2012}
B. Steinberg,
Representation theory of finite groups.
An introductory approach, Universitext, Springer, New York, 2012.
%%%%%%%%%%%%%%%%%%%%%
\bibitem{szi2008}
\href{https://doi.org/10.1016/j.ffa.2007.09.004}
{P. Sziklai,
A bound on the number of points of a plane curve,
Finite Fields Appl. 14 (2008) 41--43.}
%%%%%%%%%%%%%%%%%%
