Algebraic Curves over Finite Fields

This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa’s construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve o...

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Detalles Bibliográficos
Autor: Rovi, Carmen
Tipo de recurso: tesis de maestría
Fecha de publicación:2010
País:España
Institución:Universidad Nacional de Educación a Distancia
Repositorio:e-spacio. Repositorio Institucional de la UNED
Idioma:inglés
OAI Identifier:oai:e-spacio.uned.es:20.500.14468/21447
Acceso en línea:https://hdl.handle.net/20.500.14468/21447
Access Level:acceso abierto
Palabra clave:12 Matemáticas
Nullstellensatz
variety
rational function
Function field
Weierstrass gap Theorem
Ramification
Hurwitz genus formula
Kummer and Artin-Schreier extensions
Hasse- Weil bound
Goppa codes
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spelling Algebraic Curves over Finite FieldsRovi, Carmen12 MatemáticasNullstellensatzvarietyrational functionFunction fieldWeierstrass gap TheoremRamificationHurwitz genus formulaKummer and Artin-Schreier extensionsHasse- Weil boundGoppa codesThis thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa’s construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to find examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/˜geer, which to the time of writing this Thesis appear as ”no information available”. In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.Universidad Nacional de Educación a Distancia (España). Facultad de CienciasIzquierdo Barrios, Mª de los Milagrose-Spacio UNED20242024-05-2120102010-06-0120102010-06-01master thesishttp://purl.org/coar/resource_type/c_bdccinfo:eu-repo/semantics/masterThesisapplication/pdfhttps://hdl.handle.net/20.500.14468/21447reponame:e-spacio. Repositorio Institucional de la UNEDinstname:Universidad Nacional de Educación a DistanciaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.esoai:e-spacio.uned.es:20.500.14468/214472026-06-06T12:38:31Z
dc.title.none.fl_str_mv Algebraic Curves over Finite Fields
title Algebraic Curves over Finite Fields
spellingShingle Algebraic Curves over Finite Fields
Rovi, Carmen
12 Matemáticas
Nullstellensatz
variety
rational function
Function field
Weierstrass gap Theorem
Ramification
Hurwitz genus formula
Kummer and Artin-Schreier extensions
Hasse- Weil bound
Goppa codes
title_short Algebraic Curves over Finite Fields
title_full Algebraic Curves over Finite Fields
title_fullStr Algebraic Curves over Finite Fields
title_full_unstemmed Algebraic Curves over Finite Fields
title_sort Algebraic Curves over Finite Fields
dc.creator.none.fl_str_mv Rovi, Carmen
author Rovi, Carmen
author_facet Rovi, Carmen
author_role author
dc.contributor.none.fl_str_mv Izquierdo Barrios, Mª de los Milagros
e-Spacio UNED
dc.subject.none.fl_str_mv 12 Matemáticas
Nullstellensatz
variety
rational function
Function field
Weierstrass gap Theorem
Ramification
Hurwitz genus formula
Kummer and Artin-Schreier extensions
Hasse- Weil bound
Goppa codes
topic 12 Matemáticas
Nullstellensatz
variety
rational function
Function field
Weierstrass gap Theorem
Ramification
Hurwitz genus formula
Kummer and Artin-Schreier extensions
Hasse- Weil bound
Goppa codes
description This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa’s construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to find examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/˜geer, which to the time of writing this Thesis appear as ”no information available”. In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.
publishDate 2010
dc.date.none.fl_str_mv 2010
2010-06-01
2010
2010-06-01
2024
2024-05-21
dc.type.none.fl_str_mv master thesis
http://purl.org/coar/resource_type/c_bdcc
dc.type.openaire.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14468/21447
url https://hdl.handle.net/20.500.14468/21447
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias
publisher.none.fl_str_mv Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias
dc.source.none.fl_str_mv reponame:e-spacio. Repositorio Institucional de la UNED
instname:Universidad Nacional de Educación a Distancia
instname_str Universidad Nacional de Educación a Distancia
reponame_str e-spacio. Repositorio Institucional de la UNED
collection e-spacio. Repositorio Institucional de la UNED
repository.name.fl_str_mv
repository.mail.fl_str_mv
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