Absolute reconstruction of number fields from the Deligne-Ribet monoids

TL;DR

This paper demonstrates that the structure of certain number fields can be uniquely reconstructed from associated algebraic monoids, extending previous results to more general cases.

Contribution

It proves the reconstruction of number fields from Deligne-Ribet monoids for specific fields and discusses extensions to the general case.

Findings

Number fields can be reconstructed from Deligne-Ribet monoids.

Reconstruction is proven for rational and imaginary quadratic fields.

Discussion on extending the reconstruction to more general number fields.

Abstract

Following Cornelissen, Li, Marcolli, and Smit, this short paper proves that the field structure of a number field $K$ can be reconstructed from the pair $(DR_K, I_K)$ of the Deligne-Ribet monoid $DR_K$ and the submonoid $I_K$ of $DR_K$, when $K$ is the rational number field, or an imaginary quadratic field. The general-case reconstruction is also discussed, which is more abstract than the case of rational and imaginary quadratic fields.