We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over or a quadratic number field. We accomplish this by studying the role of -adic heights in explicit non-Abelian Chabauty.
"Quadratic Chabauty and rational points, I: -adic heights." Duke Math. J. 167 (11) 1981 - 2038, 15 August 2018. https://doi.org/10.1215/00127094-2018-0013