Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions
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arXiv
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| Natura: | Preprint |
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2024
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| author | Elmanto, Elden Zhang, Ningchuan |
| author_facet | Elmanto, Elden Zhang, Ningchuan |
| contents | In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our generalization extends the scope of this conjecture to Artin $L$-functions of Galois representations of finite, function, and totally real number fields. The statement of this conjecture relates norms of the special values of these $L$-functions to sizes of equivariant algebraic $\mathrm{K}$-groups with coefficients in an equivariant Moore spectrum attached to a Galois representation.
We prove this conjecture in many cases, integrally, except up to a possible factor of powers of $2$ in the non-abelian and totally real number field case. In the finite field case, we further determine the group structures of their equivariant algebraic $\mathrm{K}$-groups with coefficients in Galois representations. At heart, our method lifts the Möbius inversion formula for factorizations of zeta functions as a product of $L$-functions, to the $E_1$-page of an equivariant spectral sequence converging to equivariant algebraic $\mathrm{K}$-groups. Additionally, the spectral Mackey functor structure on equivariant $\mathrm{K}$-theory allows us to incorporate certain ramified extensions that appear in these $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions Elmanto, Elden Zhang, Ningchuan K-Theory and Homology Algebraic Geometry Algebraic Topology Number Theory 19F27, 19L47, 55P91 In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our generalization extends the scope of this conjecture to Artin $L$-functions of Galois representations of finite, function, and totally real number fields. The statement of this conjecture relates norms of the special values of these $L$-functions to sizes of equivariant algebraic $\mathrm{K}$-groups with coefficients in an equivariant Moore spectrum attached to a Galois representation. We prove this conjecture in many cases, integrally, except up to a possible factor of powers of $2$ in the non-abelian and totally real number field case. In the finite field case, we further determine the group structures of their equivariant algebraic $\mathrm{K}$-groups with coefficients in Galois representations. At heart, our method lifts the Möbius inversion formula for factorizations of zeta functions as a product of $L$-functions, to the $E_1$-page of an equivariant spectral sequence converging to equivariant algebraic $\mathrm{K}$-groups. Additionally, the spectral Mackey functor structure on equivariant $\mathrm{K}$-theory allows us to incorporate certain ramified extensions that appear in these $L$-functions. |
| title | Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology Number Theory 19F27, 19L47, 55P91 |
| url | https://arxiv.org/abs/2405.03578 |