{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/105797"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/105797","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Tamely ramified geometric Langlands correspondence in positive characteristic","abstract":"We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k), where k is an algebraically closed field of characteristic p > n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bun(n,P) the moduli stack of (quasi-)parabolic vector bundles on X, and by Loc(n, P) the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category D^b(QCoh(Loc^0(n,P))) of quasi-coherent sheaves on an open substack Loc^0(n, P) of Loc(n,P), and the bounded derived category D^b(D^0Bun(n,P)-mod) of D^0Bun(n,P)-modules, where D^0Bun(n,P) is a localization of DBun(n,P) the sheaf of crystalline differential operators on Bun(n,P). Thus we extend the work of Bezrukavnikov-Braverman [8] to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles.","abstract_html":"We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k), where k is an algebraically closed field of characteristic p > n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bun(n,P) the moduli stack of (quasi-)parabolic vector bundles on X, and by Loc(n, P) the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category D^b(QCoh(Loc^0(n,P))) of quasi-coherent sheaves on an open substack Loc^0(n, P) of Loc(n,P), and the bounded derived category D^b(D^0Bun(n,P)-mod) of D^0Bun(n,P)-modules, where D^0Bun(n,P) is a localization of DBun(n,P) the sheaf of crystalline differential operators on Bun(n,P). Thus we extend the work of Bezrukavnikov-Braverman [8] to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles.","abstract_has_math":false,"creators":["Shen, Shiyu"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Nevins, Thomas","Haboush, William","McGerty, Kevin","Dodd, Christopher"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-11-26T20:49:23Z","date_published":"2019-11-26T20:49:23Z","updated_at":"2026-07-22T22:24:45Z","subjects":["Hitchin systems, geometric Langlands correspondence"],"languages":["en"],"rights":["2019 by Shiyu Shen. All rights reserved."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/105797","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nevins, Thomas","Haboush, William","McGerty, Kevin","Dodd, Christopher"]},{"key":"dc:creator","label":"Author","values":["Shen, Shiyu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-11-26T20:49:23Z","2021-11-27T10:15:23Z","2019-07-09","2019-08"]},{"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":["Hitchin systems, geometric Langlands correspondence"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["2019 by Shiyu Shen. All rights reserved."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/105797"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k), where k is an algebraically closed field of characteristic p > n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bun(n,P) the moduli stack of (quasi-)parabolic vector bundles on X, and by Loc(n, P) the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category D^b(QCoh(Loc^0(n,P))) of quasi-coherent sheaves on an open substack Loc^0(n, P) of Loc(n,P), and the bounded derived category D^b(D^0Bun(n,P)-mod) of D^0Bun(n,P)-modules, where D^0Bun(n,P) is a localization of DBun(n,P) the sheaf of crystalline differential operators on Bun(n,P). Thus we extend the work of Bezrukavnikov-Braverman [8] to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2021-08-01","The student, Shiyu Shen, accepted the attached license on 2019-07-09 at 07:07.","The student, Shiyu Shen, submitted this Dissertation for approval on 2019-07-09 at 07:24.","This Dissertation was approved for publication on 2019-07-09 at 13:48.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14211 on 2019-11-26 at 13:04:44","Made available in DSpace on 2019-11-26T20:49:23Z (GMT). No. of bitstreams: 2 SHEN-DISSERTATION-2019.pdf: 547016 bytes, checksum: 58e2e26289834f94a5d9ff8697327407 (MD5) LICENSE.txt: 4207 bytes, checksum: 669682ccb860115616cc33008b43775e (MD5) Previous issue date: 2019-07-09","Embargo set by: Seth Robbins for item 112942 Lift date: 2021-11-26T20:49:41Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 112942 on 2021-11-27T10:15:23Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Tamely ramified geometric Langlands correspondence in positive characteristic"]}]}],"canonical_facts":{"dc:contributor":["Nevins, Thomas","Haboush, William","McGerty, Kevin","Dodd, Christopher"],"dc:creator":["Shen, Shiyu"],"dc:date":["2019-11-26T20:49:23Z","2021-11-27T10:15:23Z","2019-07-09","2019-08"],"dc:description":["We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k), where k is an algebraically closed field of characteristic p > n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bun(n,P) the moduli stack of (quasi-)parabolic vector bundles on X, and by Loc(n, P) the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category D^b(QCoh(Loc^0(n,P))) of quasi-coherent sheaves on an open substack Loc^0(n, P) of Loc(n,P), and the bounded derived category D^b(D^0Bun(n,P)-mod) of D^0Bun(n,P)-modules, where D^0Bun(n,P) is a localization of DBun(n,P) the sheaf of crystalline differential operators on Bun(n,P). Thus we extend the work of Bezrukavnikov-Braverman [8] to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2021-08-01","The student, Shiyu Shen, accepted the attached license on 2019-07-09 at 07:07.","The student, Shiyu Shen, submitted this Dissertation for approval on 2019-07-09 at 07:24.","This Dissertation was approved for publication on 2019-07-09 at 13:48.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14211 on 2019-11-26 at 13:04:44","Made available in DSpace on 2019-11-26T20:49:23Z (GMT). No. of bitstreams: 2 SHEN-DISSERTATION-2019.pdf: 547016 bytes, checksum: 58e2e26289834f94a5d9ff8697327407 (MD5) LICENSE.txt: 4207 bytes, checksum: 669682ccb860115616cc33008b43775e (MD5) Previous issue date: 2019-07-09","Embargo set by: Seth Robbins for item 112942 Lift date: 2021-11-26T20:49:41Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 112942 on 2021-11-27T10:15:23Z."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/105797"],"dc:language":["en"],"dc:rights":["2019 by Shiyu Shen. All rights reserved."],"dc:subject":["Hitchin systems, geometric Langlands correspondence"],"dc:title":["Tamely ramified geometric Langlands correspondence in positive characteristic"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:45Z"}