Unlikely intersections and the Chabauty–Kim method over number fields

Abstract The Chabauty–Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental p -adic analytic functions arising from Selmer schemes associated to unipotent fundamental group of variety. In this paper we establish several foundational results curves fields. two main ingredients in proof these are an unlikely intersection result zeroes iterated integrals, and careful analysis scheme original curve with Albanese variety $${\mathbb {Q}}_p $$ Q p -subvarieties restriction scalars curve. theorem also gives partial answer question Siksek Chabauty’s fields, explicit counterexample given strong form Siksek’s question.