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arXiv · 2007.15078

The Galois action on symplectic $K$-theory

Tony Feng·Soren Galatius·Akshay Venkatesh
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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.

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Tony Feng, Soren Galatius, Akshay Venkatesh. 2020-07-29. The Galois action on symplectic $K$-theory. https://doi.org/10.1007/s00222-022-01127-8

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