On the higher algebraic $K$-groups of arithmetically equivalent number fields

Abstract

In this paper, based on the structure of higher algebraic $K$-groups of the rings of integers of number fields, we introduce new equivalence relations between number fields called $K$-equivalence and almost $K$-equivalence, and investigate their relationships with arithmetical equivalence and local integral equivalence. Building upon the classical properties of arithmetically equivalent number fields studied by R. Perlis and on the pioneering work by Komatsu, we apply the Rost-Voevodsky theorem (the Quillen-Lichtenbaum conjecture) for odd primes $p$, thereby analyzing algebraic $K$-groups within the framework of continuous étale cohomology from a more modern perspective. As our main results, by utilizing permutation representations of global Galois groups and the data of local decomposition groups, we refine Komatsu's previous results in the range $p \neq 2$ and describe the conditions for number fields to be (almost) $K$-equivalent. Through these developments, we clarify how the arithmetic information reflected in higher $K$-groups resonates with the special values of zeta functions and Galois representations.