Functional transcendence for the unipotent Albanese map

TL;DR

This paper establishes a transcendence property of the unipotent Albanese map under the Ax-Schanuel conjecture, enabling the extension of Chabauty-Kim Diophantine methods to higher-dimensional varieties and broader number fields.

Contribution

It introduces a conditional transcendence result that generalizes Chabauty-Kim techniques to higher dimensions and arbitrary number fields.

Findings

Conditional proof of transcendence property of the unipotent Albanese map

Extension of Chabauty-Kim method to higher-dimensional varieties

Generalization of Diophantine finiteness results to arbitrary number fields

Abstract

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields.