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.

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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