L-functions and J-spectra

Zhang, Ningchuan

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https://hdl.handle.net/2142/107887 Copy

Description

Title

L-functions and J-spectra

Author(s)

Zhang, Ningchuan

Issue Date

2020-04-23

Director of Research (if dissertation) or Advisor (if thesis)

Ando, Matthew

Doctoral Committee Chair(s)

Rezk, Charles

Committee Member(s)

• Allen, Patrick
• Stojanoska, Vesna

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Chromatic homotopy theory
• J-spectra
• Dirichlet L-functions
• Eisenstein series

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.

Graduation Semester

2020-05

Type of Resource

Thesis

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http://hdl.handle.net/2142/107887

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© 2020 by Ningchuan Zhang. All rights reserved.

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