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arXiv · 2209.10044

Borel's rank theorem for Artin $L$-functions

Ningchuan Zhang
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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.

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BibTeXRIS

Ningchuan Zhang. 2022-09-20. Borel's rank theorem for Artin $L$-functions. https://doi.org/10.1090/proc/16493

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