Vol. 15, No. 6, 2021

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Functional transcendence for the unipotent Albanese map

Daniel Rayor Hast

Vol. 15 (2021), No. 6, 1565–1580
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.

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