Globally valued fields: foundations

TL;DR

This paper develops the foundational theory of globally valued fields, linking model theory, Arakelov geometry, and measure theory, and establishes a representation theorem connecting these fields with adelic curves.

Contribution

It introduces the concept of globally valued fields, providing a unifying framework and a representation theorem relating them to adelic curves.

Findings

Established a dictionary between different data defining globally valued fields

Proved a representation theorem relating globally valued fields to adelic curves

Unified various perspectives on the structure of global fields

Abstract

We present foundations of globally valued fields, i.e., of a class of fields with an extra structure, capturing some aspects of the geometry of global fields, based on the product formula. We provide a dictionary between various data defining such extra structure: syntactic (models of some unbounded continuous logic theory), Arakelov theoretic, and measure theoretic. In particular we obtain a representation theorem relating globally valued fields and adelic curves defined by Chen and Moriwaki.