Embedded Projective Curves over a Finite Field and Homma Constant $D(q)$

Yıl 2022, Cilt: 4 Sayı: 1, 17 - 19, 24.07.2022

Edoardo Ballico

Öz

We consider the existence of smooth projective curves embedded over a fixed finite field $\mathbb{F}_q$ and such that their ratio $\#X(\mathbb {F}_q)/\deg(X)$ is large. We discuss the geometry of curves computing the Iihara constants $A(q)$ and $A^-(q)$ and relate it to upper and lower bound of the Homma constants $D(q)$ and $D^-(q)$ .

Anahtar Kelimeler

Finite field, curve over a finite field, curves in projective spaces

Kaynakça

• Niederreiter, H., & Xing, C. (2001). Rational points on curves over finite fields: theory and applications, Cambridge University Press, Cambridge.
• Niederreiter, H., & Xing, C. (2009). Algebraic Geometry in Coding Theory and Cryptography, Princeton University Press, Princeton, NJ.
• Serre, J. P., Howe, E. W., Oesterlé, J., & Ritzenthaler, C. (2020). Rational points on curves over finite fields, Documents Mathématiques, 18, Société Mathématique de France, Paris.
• Stichtenoth, H. (2009). Algebraic function fields and codes, Second Edition. Springer-Verlag.
• Tsfasman, M., Vlădut, S., & Nogin, D. (2007). Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, 139.
• Homma, M. (2012). A bound on the number of points of a curve in a projective space over a finite field, Theory and Applications of Finite Fields, 597, 103-110.
• Beelen, P., Montanucci, M., & Vicino, L. (2022). On the constant D(q) defined by Homma. arXiv:2201.00602; accepted in Proceedings of the 18th Conference on Arithmetic, Geometry, Cryptography, and Coding Theory in the AMS book series Contemporary Mathematics (CONM).
• Beelen, P., & Montanucci, M. (2020). A bound for the number of points of space curves over finite fields. arXiv:2008.05748.