2026-09-21T17:41:50Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/107887 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Ando, Matthew Rezk, Charles Allen, Patrick Stojanoska, Vesna Zhang, Ningchuan 2020-08-26T21:54:20Z 2020-08-26T21:54:20Z 2020-04-23 2020-05 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). No. of bitstreams: 4 ZHANG-DISSERTATION-2020.pdf: 641023 bytes, checksum: 2fd006f6b04f4ff8329da40330bf87d0 (MD5) Latex source code v2.zip: 129450 bytes, checksum: d62af2f7c183ef4d216d65712ae9c553 (MD5) LICENSE.txt: 4212 bytes, checksum: a1c6a616a9aa83aa4c551fb9bbbbf096 (MD5) PROQUEST_LICENSE.txt: 4558 bytes, checksum: a45ae5bd706d7ef23e1c40fcc5f8d356 (MD5) Previous issue date: 2020-04-23 application/pdf http://hdl.handle.net/2142/107887 en © 2020 by Ningchuan Zhang. All rights reserved. Chromatic homotopy theory J-spectra Dirichlet L-functions Eisenstein series L-functions and J-spectra text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.