Borel’s rank theorem for Artin $L$-functions
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- by Ningchuan Zhang
- Proc. Amer. Math. Soc. 151 (2023), 4621-4632
- DOI: https://doi.org/10.1090/proc/16493
- Published electronically: June 30, 2023
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Abstract:
Borel’s rank theorem identifies the ranks of algebraic $K$-groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin $L$-functions by considering equivariant algebraic $K$-groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic $K$-theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in $L$-functions.References
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Bibliographic Information
- Ningchuan Zhang
- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- MR Author ID: 1495949
- ORCID: 0000-0002-7003-6354
- Email: nczhang@sas.upenn.edu
- Received by editor(s): September 26, 2022
- Received by editor(s) in revised form: March 13, 2023, and March 22, 2023
- Published electronically: June 30, 2023
- Communicated by: Julie Bergner
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4621-4632
- MSC (2020): Primary 19F27; Secondary 55P62, 55P91
- DOI: https://doi.org/10.1090/proc/16493
- MathSciNet review: 4634868