On Weierstrass points and some properties of curves of Hurwitz type

Author(s)

Grégory Duran Cunha

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2018-02-07

Examining board members

Herivelto Martins Borges Filho; Nazar Arakelian; Cícero Fernandes de Carvalho; Behrooz Mirzaii

• Advisor: Herivelto Martins Borges Filho

Abstract

This work presents several results on curves of Hurwitz type, defined over a finite field. In 1961, Tallini investigated plane irreducible curves of minimum degree containing all points of the projective plane PG(2,q) over a finite field of order q. We prove that such curves are Fq3(q2+q+1)-projectively equivalent to the Hurwitz curve of degree q+2, and compute some of itsWeierstrass points. In addition, we prove that when q is prime the curve is ordinary, that is, the p-rank equals the genus of the curve. We also compute the automorphism group of such curve and show that some of the quotient curves, arising from some special cyclic automorphism groups, are still curves of Hurwitz type. Furthermore, we solve the problem of explicitly describing the set of all Weierstrass pure gaps supported by two or three special points on Hurwitz curves. Finally, we use the latter characterization to construct Goppa codes with good parameters, some of which are current records in the Mint table. (AU)

• FAPESP's process: 14/03366-9 - The Galois closure of the multi-Frobenius nonclassical curves
• Grantee: Grégory Duran Cunha
• Support Opportunities: Scholarships in Brazil - Doctorate