On plane maximal curves
A. Cossidente, J.W.P. Hirschfeld, G. Korchmaros, F. Torres

TL;DR
This paper characterizes a specific class of maximal curves over finite fields with r^2 elements, identifying them as Fermat curves of a certain degree, expanding understanding of their geometric and algebraic properties.
Contribution
It introduces a new characterization of maximal curves with genus g_2 as Fermat curves, filling a gap in the classification of such curves.
Findings
Maximal curves with genus g_2 are Fermat curves of degree (r+1)/2.
Characterization applies to curves with a non-singular plane model.
Extends classification beyond previously known genus cases.
Abstract
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2.
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Taxonomy
TopicsCoding theory and cryptography · Algebraic Geometry and Number Theory · Finite Group Theory Research
