Detecting the $\beta$-family in iterated algebraic K-theory of finite fields
Gabriel Angelini‐Knoll · arXiv (Cornell University) · 2018
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 prove a highly nontrivial case this conjecture. 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 $R=K(\mathbb{F}_q)_p$ where $p\ge 5$ and $q$ is prime power generator of the units in $\mathbb{Z}/p^2\mathbb{Z}$. In particular, we prove that the commutative ring spectrum $K(K(\mathbb{F}_q)_p)$ detects the $\beta$-family. The method of proof also implies that the $\beta$-family is detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to modular forms satisfying certain congruences.