{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101037"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101037","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Covers and invariants of Deligne-Lusztig curves","abstract":"This thesis is comprised of three parts, each dealing with one or more of the Hermitian, Suzuki, and Ree curves, which are three families of algebraic curves over finite fields which have pronounced arithmetic and geometric properties. In the first part, we use a ray class field construction to produce covers of each of these three families of curves which meet the Hasse-Weil bound over suitable base fields. In the Hermitian case, the family of covers constructed coincide with the family of Giulietti-Korchmáros curves. In the second part, we study a certain linear series D on the Ree curve which gives an embedding in P^13. We compute the orders of vanishing of sections of D, and use this to determine the set of Weierstrass points of D. The third part is a computational project studying the structure of the 3-torsion group scheme of the Jacobian of the smallest Ree curve, which has genus 3627. This is accomplished by computing the action of the Frobenius and Verschiebung operators on the de Rham cohomology of the curve. As a result, we determine the Ekedahl-Oort type and a decomposition for the Dieudonné module of this curve.","abstract_html":"This thesis is comprised of three parts, each dealing with one or more of the Hermitian, Suzuki, and Ree curves, which are three families of algebraic curves over finite fields which have pronounced arithmetic and geometric properties. In the first part, we use a ray class field construction to produce covers of each of these three families of curves which meet the Hasse-Weil bound over suitable base fields. In the Hermitian case, the family of covers constructed coincide with the family of Giulietti-Korchmáros curves. In the second part, we study a certain linear series D on the Ree curve which gives an embedding in P^13. We compute the orders of vanishing of sections of D, and use this to determine the set of Weierstrass points of D. The third part is a computational project studying the structure of the 3-torsion group scheme of the Jacobian of the smallest Ree curve, which has genus 3627. This is accomplished by computing the action of the Frobenius and Verschiebung operators on the de Rham cohomology of the curve. As a result, we determine the Ekedahl-Oort type and a decomposition for the Dieudonné module of this curve.","abstract_has_math":false,"creators":["Skabelund, Dane C."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Duursma, Iwan","Ahlgren, Scott","Nevins, Thomas","Allen, Patrick"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-04T20:27:22Z","date_published":"2018-09-04T20:27:22Z","updated_at":"2026-07-22T22:24:38Z","subjects":["Deligne-Lusztig curves","Ree curve","maximal curves","supersingular curves","Stohr-Voloch theory","Weierstrass points","p-torsion","de Rham cohomology"],"languages":["en"],"rights":["Copyright 2018 Dane Skabelund"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101037","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Duursma, Iwan","Ahlgren, Scott","Nevins, Thomas","Allen, Patrick"]},{"key":"dc:creator","label":"Author","values":["Skabelund, Dane C."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-04T20:27:22Z","2018-04-20","2018-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Deligne-Lusztig curves","Ree curve","maximal curves","supersingular curves","Stohr-Voloch theory","Weierstrass points","p-torsion","de Rham cohomology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Dane Skabelund"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101037"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is comprised of three parts, each dealing with one or more of the Hermitian, Suzuki, and Ree curves, which are three families of algebraic curves over finite fields which have pronounced arithmetic and geometric properties. In the first part, we use a ray class field construction to produce covers of each of these three families of curves which meet the Hasse-Weil bound over suitable base fields. In the Hermitian case, the family of covers constructed coincide with the family of Giulietti-Korchmáros curves. In the second part, we study a certain linear series D on the Ree curve which gives an embedding in P^13. We compute the orders of vanishing of sections of D, and use this to determine the set of Weierstrass points of D. The third part is a computational project studying the structure of the 3-torsion group scheme of the Jacobian of the smallest Ree curve, which has genus 3627. This is accomplished by computing the action of the Frobenius and Verschiebung operators on the de Rham cohomology of the curve. As a result, we determine the Ekedahl-Oort type and a decomposition for the Dieudonné module of this curve.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-08-31 without embargo terms","The student, Dane Skabelund, accepted the attached license on 2018-04-20 at 12:50.","The student, Dane Skabelund, submitted this Dissertation for approval on 2018-04-20 at 13:12.","This Dissertation was approved for publication on 2018-04-20 at 13:48.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12405 on 2018-08-31 at 17:14:02","Made available in DSpace on 2018-09-04T20:27:22Z (GMT). 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In the first part, we use a ray class field construction to produce covers of each of these three families of curves which meet the Hasse-Weil bound over suitable base fields. In the Hermitian case, the family of covers constructed coincide with the family of Giulietti-Korchmáros curves. In the second part, we study a certain linear series D on the Ree curve which gives an embedding in P^13. We compute the orders of vanishing of sections of D, and use this to determine the set of Weierstrass points of D. The third part is a computational project studying the structure of the 3-torsion group scheme of the Jacobian of the smallest Ree curve, which has genus 3627. This is accomplished by computing the action of the Frobenius and Verschiebung operators on the de Rham cohomology of the curve. As a result, we determine the Ekedahl-Oort type and a decomposition for the Dieudonné module of this curve.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-08-31 without embargo terms","The student, Dane Skabelund, accepted the attached license on 2018-04-20 at 12:50.","The student, Dane Skabelund, submitted this Dissertation for approval on 2018-04-20 at 13:12.","This Dissertation was approved for publication on 2018-04-20 at 13:48.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12405 on 2018-08-31 at 17:14:02","Made available in DSpace on 2018-09-04T20:27:22Z (GMT). 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