{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107887"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107887","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"L-functions and J-spectra","abstract":"The relation between Eisenstein series and the J-homomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζ-function. Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by Dirichlet characters. We first explain congruences of these twisted Eisenstein series of level Γ_1(N) and character χ via the Dieudonné theory of height 1 formal groups and formal A-modules and their finite subgroups. Our approach is based on Katz’s algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E_2k of level 1. The crucial step is to translate the Dirichlet character χ to the Galois descent data of formal A-modules. We further connect congruences of modular forms in the Eisenstein subspace E_k(Γ_1(N),χ) with certain group cohomology involving the Dirichlet character χ. When χ is trivial, this group cohomology is on the E_2-page of a spectral sequence to compute homotopy groups of the K(1)-local sphere, which is the p-completion of the J-spectra. This gives a new explanation of the connection between congruences of E_2k and the image of the stable J-homomorphism in the stable homotopy groups of spheres. Following our analysis of congruences of Eisenstein series, we introduce the Dirichlet J-spectra. The homotopy groups of the Dirichlet J-spectra are related to the special values of the Dirichlet L-functions, and thus to congruences of the twisted Eisenstein series. Moreover, the pattern of these homotopy groups suggests a possible Brown-Comenetz duality of the Dirichlet J-spectra, which resembles the functional equations of the Dirichlet L-functions. In this sense, the Dirichlet J-spectra constructed in this paper are analogs of Dirichlet L-functions in chromatic homotopy theory.","abstract_html":"The relation between Eisenstein series and the J-homomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζ-function. Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by Dirichlet characters. We first explain congruences of these twisted Eisenstein series of level Γ_1(N) and character χ via the Dieudonné theory of height 1 formal groups and formal A-modules and their finite subgroups. Our approach is based on Katz’s algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E_2k of level 1. The crucial step is to translate the Dirichlet character χ to the Galois descent data of formal A-modules. We further connect congruences of modular forms in the Eisenstein subspace E_k(Γ_1(N),χ) with certain group cohomology involving the Dirichlet character χ. When χ is trivial, this group cohomology is on the E_2-page of a spectral sequence to compute homotopy groups of the K(1)-local sphere, which is the p-completion of the J-spectra. This gives a new explanation of the connection between congruences of E_2k and the image of the stable J-homomorphism in the stable homotopy groups of spheres. Following our analysis of congruences of Eisenstein series, we introduce the Dirichlet J-spectra. The homotopy groups of the Dirichlet J-spectra are related to the special values of the Dirichlet L-functions, and thus to congruences of the twisted Eisenstein series. Moreover, the pattern of these homotopy groups suggests a possible Brown-Comenetz duality of the Dirichlet J-spectra, which resembles the functional equations of the Dirichlet L-functions. In this sense, the Dirichlet J-spectra constructed in this paper are analogs of Dirichlet L-functions in chromatic homotopy theory.","abstract_has_math":false,"creators":["Zhang, Ningchuan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ando, Matthew","Rezk, Charles","Allen, Patrick","Stojanoska, Vesna"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:20Z","date_published":"2020-08-26T21:54:20Z","updated_at":"2026-07-22T22:24:47Z","subjects":["Chromatic homotopy theory","J-spectra","Dirichlet L-functions","Eisenstein series"],"languages":["en"],"rights":["© 2020 by Ningchuan Zhang. All rights reserved."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107887","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ando, Matthew","Rezk, Charles","Allen, Patrick","Stojanoska, Vesna"]},{"key":"dc:creator","label":"Author","values":["Zhang, Ningchuan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:20Z","2020-04-23","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Chromatic homotopy theory","J-spectra","Dirichlet L-functions","Eisenstein series"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["© 2020 by Ningchuan Zhang. All rights reserved."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107887"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The relation between Eisenstein series and the J-homomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζ-function. Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by Dirichlet characters. We first explain congruences of these twisted Eisenstein series of level Γ_1(N) and character χ via the Dieudonné theory of height 1 formal groups and formal A-modules and their finite subgroups. Our approach is based on Katz’s algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E_2k of level 1. The crucial step is to translate the Dirichlet character χ to the Galois descent data of formal A-modules. We further connect congruences of modular forms in the Eisenstein subspace E_k(Γ_1(N),χ) with certain group cohomology involving the Dirichlet character χ. When χ is trivial, this group cohomology is on the E_2-page of a spectral sequence to compute homotopy groups of the K(1)-local sphere, which is the p-completion of the J-spectra. This gives a new explanation of the connection between congruences of E_2k and the image of the stable J-homomorphism in the stable homotopy groups of spheres. Following our analysis of congruences of Eisenstein series, we introduce the Dirichlet J-spectra. The homotopy groups of the Dirichlet J-spectra are related to the special values of the Dirichlet L-functions, and thus to congruences of the twisted Eisenstein series. Moreover, the pattern of these homotopy groups suggests a possible Brown-Comenetz duality of the Dirichlet J-spectra, which resembles the functional equations of the Dirichlet L-functions. In this sense, the Dirichlet J-spectra constructed in this paper are analogs of Dirichlet L-functions in chromatic homotopy theory.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Ningchuan Zhang, accepted the attached license on 2020-04-15 at 11:11.","The student, Ningchuan Zhang, submitted this Dissertation for approval on 2020-04-15 at 12:04.","This Dissertation was approved for publication on 2020-04-23 at 14:37.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14983 on 2020-08-25 at 17:07:02","Made available in DSpace on 2020-08-26T21:54:20Z (GMT). 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Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by Dirichlet characters. We first explain congruences of these twisted Eisenstein series of level Γ_1(N) and character χ via the Dieudonné theory of height 1 formal groups and formal A-modules and their finite subgroups. Our approach is based on Katz’s algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E_2k of level 1. The crucial step is to translate the Dirichlet character χ to the Galois descent data of formal A-modules. We further connect congruences of modular forms in the Eisenstein subspace E_k(Γ_1(N),χ) with certain group cohomology involving the Dirichlet character χ. When χ is trivial, this group cohomology is on the E_2-page of a spectral sequence to compute homotopy groups of the K(1)-local sphere, which is the p-completion of the J-spectra. This gives a new explanation of the connection between congruences of E_2k and the image of the stable J-homomorphism in the stable homotopy groups of spheres. Following our analysis of congruences of Eisenstein series, we introduce the Dirichlet J-spectra. The homotopy groups of the Dirichlet J-spectra are related to the special values of the Dirichlet L-functions, and thus to congruences of the twisted Eisenstein series. Moreover, the pattern of these homotopy groups suggests a possible Brown-Comenetz duality of the Dirichlet J-spectra, which resembles the functional equations of the Dirichlet L-functions. In this sense, the Dirichlet J-spectra constructed in this paper are analogs of Dirichlet L-functions in chromatic homotopy theory.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Ningchuan Zhang, accepted the attached license on 2020-04-15 at 11:11.","The student, Ningchuan Zhang, submitted this Dissertation for approval on 2020-04-15 at 12:04.","This Dissertation was approved for publication on 2020-04-23 at 14:37.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14983 on 2020-08-25 at 17:07:02","Made available in DSpace on 2020-08-26T21:54:20Z (GMT). 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