ISSN:
1570-5846
Keywords:
finite field
;
maximal curve
;
linear series
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The number N of rational points on an algebraic curve of genus g over a finite field $${\mathbb{F}}_q $$ satisfies the Hasse–Weil bound $$N \leqslant q + 1 + 1g\sqrt q $$ . A curve that attains this bound is called maximal. With $$g_0 = \frac{1}{2}(q - \sqrt q )$$ and $$g_1 = \frac{1}{4}(\sqrt q - 1)^2 $$ , it is known that maximalcurves have $$g = g_0 or g \leqslant {\text{ }}g_1 $$ . Maximal curves with $$g = g_0 or g_1 $$ have been characterized up to isomorphism. A natural genus to be studied is $$g_2 = \frac{1}{8}(\sqrt q - 1)(\sqrt q - 3),$$ and for this genus there are two non-isomorphic maximal curves known when $$\sqrt q \equiv 3 (\bmod 4)$$ . Here, a maximal curve with genus g 2 and a non-singular plane model is characterized as a Fermat curve of degree $$\frac{1}{2}(\sqrt q + 1)$$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1001826520682
Permalink
