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

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.