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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2020.tde-10062020-134739
Document
Author
Full name
Cirilo Gonçalves Júnior
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Borges Filho, Herivelto Martins (President)
Arakelian, Nazar
Carvalho, Cícero Fernandes de
Tafazolian, Saeed
 
Title in English
On Fq3 -Frobenius nonclassical curvces of type Yq2+q+1 = f (X) and the Hasse-Witt invariant for a class of Kummer extension
Keywords in English
α-number
Automorphism group
Frobenius nonclassical curves
Hasse-Witt invariant
Hyperelliptic curves
Abstract in English
Inserted in the context of algebraic curves defined over finite fields, this work presents several results in two different topics. First, it gives a complete characterization of the Fq3 -Frobenius nonclassical curves of type Yq2+q+1 = f (X), and it provides an explicit computation of the following birational invariants: genus, automorphism group, Hasse-Witt invariant and a-number. The number of Fq3 -rational points is computed as well. Second, this work provides an extensive study of the Hasse-Witt invariant of the curves Ym +Xn +1 = 0 and Ym +Xn +X = 0. A combinatorial formula for this invariant is presented in the general case, and explicit closed formulas are provided for special values of m and n.
 
Title in Portuguese
Sobre curvas Fq3 -Frobenius não-clássicas do tipo Yq2+q+1 = f (X) e o invariante de Hasse-Witt para uma classe de extensões de Kummer
Keywords in Portuguese
α-number
Curvas Frobenius não-classicas
Curvas hiperelípticas
Grupo de automorfismo
Invariante de Hasse-Witt
Abstract in Portuguese
Inserido no contexto de curvas algébricas definidas sobre corpos finitos, este trabalho apresenta vários resultados em dois tópicos diferentes. Primeiro, ele apresenta uma caracterização completa das curvas Fq3 -Frobenius não-clássicas do tipo Yq2+q+1 = f (X) e fornece um cálculo explícito dos seguintes invariantes birracionais: gênero, grupo automorfismo, invariante de Hasse-Witt e anumber. O número de pontos Fq3 -racionais também é calculado. Segundo, este trabalho fornece um extensivo estudo do invariante de Hasse-Witt das curvas Ym +Xn +1 = 0 e Ym +Xn +X = 0. Uma fórmula combinatória para esse invariante é apresentada no caso geral, e fórmulas fechadas explícitas são fornecidas para valores especiais de m e n.
 
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Publishing Date
2020-06-10
 
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