Since F. Gaßmann’s 1926 paper [Math. Z. 25, 661–675 (1926; JFM 52.0156.03)], it has been known that there exist distinct number fields with a common zeta function. The failure of these and related invariants to differentiate between number fields (and, more generally, global fields) can be addressed by means of dynamical systems. In particular, the authors prove that global fields \(\mathbb{K}, \mathbb{L}\) are isomorphic if and only if there is an orbit equivalence of certain dynamical systems induced by the Artin reciprocity map for each.
Given \(\mathbb{K}\), the action defining the dynamical system comes about from “the group of fractional ideals (finite ideles modulo idelic units \(\widehat{\mathscr O}^{*}_{\mathbb{K}}\)) and also the monoid \(I_{\mathbb{K}}\) of integral ideals, act[ing] on [...] the quotient of \(G^{\text{ab}}_{\mathbb{K}} \times \widehat{\mathscr O}_{\mathbb{K}}\) by the subgroup \(\{(\text{rec}^{-1}(u), u) \mid u \in \widehat{\mathscr O}^{*}_{\mathbb{K}}\), where \(\widehat{\mathscr O}_{\mathbb{K}}\) is the set of integral adeles” of the field \(\mathbb{K}\).
The paper is well motivated and clearly cites related work. The authors raise interesting questions, including: “Can one reconstruct a number field from its associated dynamical system ... rather than ... reconstructing an isomorphism of fields from an isomorphism of their dynamical systems ...”?