The Galois action on symplectic $K$-theory

TL;DR

This paper investigates the Galois group action on a symplectic variant of algebraic K-theory of integers, explicitly computing the representations and characterizing their universal properties using complex multiplication theory.

Contribution

It provides the first explicit computation of the Galois action on symplectic algebraic K-theory and characterizes the resulting representations as universal extensions.

Findings

Representations are extensions of Tate twists by trivial representations.

Explicit description of the Galois action on symplectic K-theory.

Characterization of these extensions via a universal property.

Abstract

We study a symplectic variant of algebraic $K$-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of $\mathbf{Q}$. We compute this action explicitly. The representations we see are extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties.