Detecting $\beta$ elements in iterated algebraic K-theory
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Abstract:
The Lichtenbaum–Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni–Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the $n$-th Greek letter family is detected by a commutative ring spectrum $R$, then we conjecture that the $n+1$-st Greek letter family will be detected by the algebraic K-theory of $R$. We prove this in the case $n=1$ for $R=\mathrm {K}(\mathbb {F}_q)$ modulo $(p,v_1)$ where $p\ge 5$ and $q=\ell ^k$ is a prime power generator of the units in $\mathbb {Z}/p^2\mathbb {Z}$. In particular, we prove that the commutative ring spectrum $\mathrm {K}(\mathrm {K}(\mathbb {F}_q))$ detects the part of the $p$-primary $\beta$-family that survives mod $(p,v_1)$. The method of proof also implies that these $\beta$ elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences.References
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Additional Information
- Gabriel Angelini-Knoll
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824; and Institut fur Mathematik, Freie Universität Berlin, Arnimallee 7, 14195 Berlin, Germany
- Address at time of publication: Department of Mathematics, Institut Galilée, Université Sorbonne Paris Nord, 99 Av. JB Clément, FR-93430 Villetaneuse, France
- MR Author ID: 1284822
- ORCID: 0000-0002-2002-4398
- Email: angelini-knoll@math.univ-paris13.fr
- Received by editor(s): November 11, 2018
- Received by editor(s) in revised form: June 7, 2021, December 10, 2021, and July 27, 2022
- Published electronically: January 12, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 2657-2692
- MSC (2020): Primary 55Q51, 19D55, 11F33, 19L20; Secondary 55P42, 55P43, 55T15, 19D50
- DOI: https://doi.org/10.1090/tran/8833
- MathSciNet review: 4557878