L-functions and J-spectra

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2020

Author dc:creator

• Zhang, Ningchuan

Contributors dc:contributor

• Ando, Matthew
• Rezk, Charles
• Allen, Patrick
• Stojanoska, Vesna

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

Statement dc:rights

• © 2020 by Ningchuan Zhang. All rights reserved.

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/107887