La logique continue des corps globalement valu\'es

TL;DR

This paper explores the model-theoretic properties of globally valued fields, including algebraic numbers, using continuous logic, and proves their existential closure in the context of diophantine equations and heights.

Contribution

It demonstrates that the field of algebraic numbers is existentially closed as a globally valued field, extending previous work to number fields using Arakelov geometry.

Findings

Algebraic numbers form an existentially closed globally valued field.

The proof adapts methods from function fields to number fields.

Connections between continuous logic and diophantine geometry are established.

Abstract

The continuous logic of globally valued fields -- A globally valued field is a field endowed with a family of absolute values that satisfy a product formula. Number fields and function fields in one variable give classical and fundamental examples; Nevanlinna theory also gives rise to such structures on the field of meromorphic functions on $\mathbf C$. These globally valued fields can be studied in the context of continuous logic (for which the predicates are real valued), and such a study has been undertaken some 10 years ago by Ben Yaacov and Hrushovski, thus providing a model-theoretic framework for the diophantine theory of heights. One of the first fundamental results in the tehory states the the field of algebraic numbers, with its essentially unique structure of a globally valued field, is existentially closed: every system involving polynomial equalities and inequalities, as well as strict inequalities in heights, possesses a solution in algebraic numbers as soon as it possesses some solution in a globally valued extension. The proof, due to Szachniewicz, is inspired by the proof proposed by Ben Yaacov and Hrushovski in the case of function fields: the latter used in a crucial way the description by Boucksom, Demailly, P\u aun and Peternell of the cone of mobile curves in a complex projective variety, the case of number fields relies on recent results in Arakelov geometry.