Detecting 𝛽 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 n -th Greek letter family is detected by a commutative ring spectrum R R , then we conjecture that the n + 1 n+1 -st Greek letter family will be detected by the algebraic K-theory of R R . We prove this in the case n = 1 n=1 for R = K ( F q ) R=\mathrm {K}(\mathbb {F}_q) modulo ( p , v 1 ) (p,v_1) where p ≥ 5 p\ge 5 and q = ℓ k q=\ell ^k is a prime power generator of the units in Z / p 2 Z \mathbb {Z}/p^2\mathbb {Z} . In particular, we prove that the commutative ring spectrum K ( K ( F q ) ) \mathrm {K}(\mathrm {K}(\mathbb {F}_q)) detects the part of the p p -primary β \beta -family that survives mod ( p , v 1 ) (p,v_1) . The method of proof also implies that these β \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.