2018/10/23 by Gabriel Angelini‐Knoll, Angelini-Knoll, Gabriel
Mathematics · #11F33 #19D50 #19D55 #55Q51 #55T15 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1810.10088
openalex publication_date 2018/10/23 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
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=K(\mathbbFq) modulo (p,v1) where p≥ 5 and q=ℓk is a prime power generator of the units in ℤ/p2ℤ. In particular, we prove that the commutative ring spectrum K(K(\mathbbFq)) detects the part of the p-primary β-family that survives mod (p,v1). The method of proof also implies that these β 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.