# On Plane Maximal Curves

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2000-04

AUTHORS ABSTRACT

The number N of rational points on an algebraic curve of genus g over a finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{F}}_q$$ \end{document} satisfies the Hasse–Weil bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$N \leqslant q + 1 + 1g\sqrt q$$ \end{document}. A curve that attains this bound is called maximal. With \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g_0 = \frac{1}{2}(q - \sqrt q )$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g_1 = \frac{1}{4}(\sqrt q - 1)^2$$ \end{document}, it is known that maximalcurves have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g = g_0 or g \leqslant {\text{ }}g_1$$ \end{document}. Maximal curves with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g = g_0 or g_1$$ \end{document} have been characterized up to isomorphism. A natural genus to be studied is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g_2 = \frac{1}{8}(\sqrt q - 1)(\sqrt q - 3),$$ \end{document} and for this genus there are two non-isomorphic maximal curves known when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt q \equiv 3 (\bmod 4)$$ \end{document}. Here, a maximal curve with genus g2 and a non-singular plane model is characterized as a Fermat curve of degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{1}{2}(\sqrt q + 1)$$ \end{document}. More... »

PAGES

163-181

### References to SciGraph publications

• 1995-12. The genus of maximal function fields over finite fields in MANUSCRIPTA MATHEMATICA
• 1987-12. Wronskians and linear independence in fields of prime characteristic in MANUSCRIPTA MATHEMATICA
• 1986-04. Weierstrass points on certain non-classical curves in ARCHIV DER MATHEMATIK
• 1999-05. On maximal curves in characteristic two in MANUSCRIPTA MATHEMATICA
• 1996-12. The genus of curves over finite fields with many rational points in MANUSCRIPTA MATHEMATICA
• ### Journal

TITLE

Compositio Mathematica

ISSUE

2

VOLUME

121

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1023/a:1001826520682

DOI

http://dx.doi.org/10.1023/a:1001826520682

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1052303248

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