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

TL;DR

This paper proposes a new higher chromatic height generalization of the Lichtenbaum--Quillen conjecture, providing evidence at height two by detecting $eta$-elements in iterated algebraic K-theory, linking it to modular forms.

Contribution

It introduces an alternative conjecture relating Greek letter families detection to algebraic K-theory, and proves detection of $eta$-elements in specific cases, advancing understanding of chromatic phenomena.

Findings

Detection of $eta$-elements in iterated algebraic K-theory of $ ext{K}( ext{K}( ext{F}_q))$

Evidence supporting the conjecture at height two

Connection between algebraic K-theory and modular forms

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.