Explicit class field theory and the algebraic geometry of $\Lambda$-rings

TL;DR

This paper explores the use of $ ext{Lambda}$-structures on algebraic objects over number rings to classify finite flat $ ext{Lambda}$-rings, connect periodic loci to abelian extensions, and propose a framework for explicit class field theory.

Contribution

It introduces a new perspective on class field theory via $ ext{Lambda}$-schemes, classifies maximal $ ext{Lambda}$-rings using Galois theory, and relates periodic Witt vectors to class field theory.

Findings

Maximal reduced finite flat $ ext{Lambda}$-rings classified by ray class monoid.

Periodic loci generate abelian extensions of number fields.

Classical explicit class field theories fit into the $ ext{Lambda}$-scheme framework.

Abstract

We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study reduced finite flat $\Lambda$-rings over $\mathit{O}_K$ and show that the maximal ones are classified in a Galois theoretic manner by the ray class monoid of Deligne and Ribet. Second, we show that the periodic loci on any $\Lambda$-scheme of finite type over $\mathit{O}_K$ generate a canonical family of abelian extensions of $K$. This raises the possibility that $\Lambda$-schemes could provide a framework for explicit class field theory, and we show that the classical explicit class field theories for the rational numbers and imaginary quadratic fields can be set naturally in this framework. This approach has the further merit of allowing for some precise questions in the spirit of Hilbert's 12th Problem. In an interlude which might be of independent interest, we define rings of periodic big Witt vectors and relate them to the global class field theoretical mathematics of the rest of the paper.