Functional transcendence for the unipotent Albanese map

Hast, Daniel Rayor

doi:10.2140/ant.2021.15.1565

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Functional transcendence for the unipotent Albanese map

Daniel Rayor Hast

Vol. 15 (2021), No. 6, 1565–1580

DOI: 10.2140/ant.2021.15.1565

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.

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Keywords

rational points on varieties, algebraic curves over number fields, nonabelian Chabauty, p-adic Ax–Schanuel, Hodge theory, unipotent Albanese map

Mathematical Subject Classification 2010

Primary: 11G25

Secondary: 14G20

Milestones

Received: 5 March 2020

Revised: 27 November 2020

Accepted: 1 January 2021

Published: 16 October 2021

Authors

Daniel Rayor Hast
Department of Mathematics & Statistics Boston University Boston, MA United States

Vol. 15, No. 6, 2021