K-theoretic Tate-Poitou duality at prime 2

TL;DR

This paper extends K-theoretic Tate-Poitou duality to the prime 2, addressing complexities from real embeddings, and identifies the homotopy type of the fiber of the cyclotomic trace at prime 2.

Contribution

It completes the duality framework at prime 2 and relates the homotopy fiber of the cyclotomic trace to algebraic K-theory of integers.

Findings

Duality at prime 2 is established with real embedding considerations.

Homotopy type of the fiber of the cyclotomic trace at prime 2 is identified.

Addresses complexities introduced by real embeddings in number fields.

Abstract

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized algebraic K-theory of the ring of $p$-integers in a number field and its completion using the $\mathbb{Z}_p$-Anderson duality. This paper completes the picture by addressing the prime 2, where the real embeddings of number fields introduce extra complexities. As an application, we identify the homotopy type at prime 2 of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of the integers.