Detecting $\beta$ elements in iterated algebraic K-theory

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 $R=\text{K}({\mathbb{F}}_q)$ 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 $\text{K}(\text{K}({\mathbb{F}}_q))$ 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.

arXiv comment33 pages including references. To appear in Trans. Am. Math. Soc. This version may differ slightly from the published version