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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Detecting $\beta$ elements in iterated algebraic K-theory
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by Gabriel Angelini-Knoll HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 2657-2692 Request permission


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.
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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:
  • 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:
  • MathSciNet review: 4557878