Quantitativity on the number of rational points in the Mordell conjecture

Abstract: In this paper, we prove an explicit upper bound on the number of rational points on a smooth projective curve of genus at least two over a number field. This gives explicit constants in the uniform Mordell conjecture proposed by Mazur and proved by Vojta, Dimitrov-Gao-Habegger, and Kühne. The main body of this paper consists of two parts: Part I for arithmetic estimates and Part II for analytic estimates.