The Galois action on symplectic K-theory

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.