Research PaperNo Access

The rank of the Cartier operator on Picard curves

Faculty of Mathematics and Computer Science, Amirkabir University of Technology, (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran

E-mail Address: nourozi@aut.ac.ir

E-mail Address: nourozi.v@gmail.com

Search for more papers by this author

and

Farhad Rahmati

https://orcid.org/0000-0002-4859-6179

Faculty of Mathematics and Computer Science, Amirkabir University of Technology, (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran

E-mail Address: frahmati@aut.ac.ir

Search for more papers by this author

https://doi.org/10.1142/S1793830924500289Cited by:0 (Source: Crossref)

Abstract

For an algebraic curve $𝒳$ defined over an algebraically closed field of characteristic $p > 0$ , $a$ -number $a (𝒳)$ is the dimension of the space of exact holomorphic differentials on $𝒳$ . We computed the $a$ -number for a family of certain Picard curves using the action of the Cartier operator on $H^{0} (𝒳, Ω^{1})$ .

Communicated by Xiao-Dong Zhang

Keywords:

AMSC: 11G20, 14H, 14G15

References

1. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265, Computational Algebra and Number Theory (Springer, London, 1993). Google Scholar
2. P. Cartier , Une nouvelle opration sur les formes diffÃ©rentielles, C. R. Acad. Sci. Paris 244 (1957) 426–428. Google Scholar
3. P. Cartier , Questions de rationalit des diviseurs en gomtrie algbrique, Bull. Soc. Math. France 86 (1958) 177–251. Crossref, Google Scholar
4. J. Estrada Sarlabous , On the Jacobian varieties of Picard curves defined over fields of characteristic $p > 0$ , Math. Nachr. 152 (1991) 392–340. Crossref, Google Scholar
5. J. Gonzlez , Hasse–Witt matrices for the Fermat curves of prime degree, Tohoku Math. J. 49 (1997) 149–163. Google Scholar
6. D. Gorenstein , An arithmetic theory of adjoint plane curves, Trans. Am. Math. Soc. 72 (1952) 414–436. Crossref, Google Scholar
7. T. Kodama and T. Washio , Hasse–Witt matrices of Fermat curves, Manuscr. Math. 60 (1988) 185–195. Crossref, Google Scholar
8. K.-Z. Li and F. Oort , Moduli of Supersingular Abelian Varieties, Lecture Notes in Mathematics, Vol. 1680 (Springer-Verlag, Berlin, 1998), iv+116pp. Crossref, Google Scholar
9. M. Montanucci and P. Speziali , The a-numbers of Fermat and Hurwitz curves, J. Pure Appl. Algebra 222 (2018) 477–488. Crossref, Google Scholar
10. V. Nourozi and F. Rahmati , The rank of the Cartier operator of the certain $F_{q}^{2}$ -Maximal function fields, Mo. J. Math. Sci. 34(2) (2022) 184–190. Google Scholar
11. V. Nourozi, F. Rahmati and S. Tafazolian , The a-number of certain hyperelliptic curves, Iran. J. Sci. Technol. Trans. A: Sci. 46(4) (2022) 1235–1239. Crossref, Google Scholar
12. V. Nourozi and S. Tafazolian , The a-number of maximal curves of the third largest genus, AUT J. Math. Comput. 3(1) (2022) 11–16. Google Scholar
13. V. Nourozi, The rank Cartier operator and linear system on curves= Classificao do operador Cartier e sistemas lineares na curva, Doctoral dissertation, (2021). Google Scholar
14. V. Nourozi, S. Tafazolian and F. Rahamti , The $a$ -number of jacobians of certain maximal curves, Trans. Comb. 10(2) (2021) 121128. Google Scholar
15. V. Nourozi, and B. Mosallaei, The $a$ -number of $y^{q^{2} + q + 1} = x^{q^{2} + 1} + x^{q}$ over finite field, arXiv, preprint (2024), arXiv:2401.13707. Google Scholar
16. R. Pries and C. Weir , The Ekedahl–Oort type of Jacobians of Hermitian curves, Asian J. Math. 19 (2015) 845–869. Crossref, Google Scholar
17. C. S. Seshadri , Lopration de Cartier. Applications, in Varits de Picard, Sminaire Claude Chevalley, Vol. 4 (Secrtariat Mathmatiques, Paris, 1958–1959). Google Scholar
18. M. Tsfasman, S. Vladut and D. Nogin , Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, Vol. 139 (American Mathematical Society, Providence, RI, 2007). Crossref, Google Scholar
19. N. Yui , On the Jacobian varieties of hyperelliptic curves over fields of characteristic p, J. Algebra 52 (1978) 378–410. Crossref, Google Scholar