Quantitativity on the number of rational points in the Mordell conjecture

TL;DR

This paper establishes explicit upper bounds on the number of rational points on high-genus curves over number fields, providing concrete constants in the uniform Mordell conjecture.

Contribution

It offers the first explicit bounds with constants for the uniform Mordell conjecture, combining arithmetic and analytic estimates.

Findings

Explicit upper bounds for rational points on curves of genus ≥ 2.

Constants in the uniform Mordell conjecture are made explicit.

The approach integrates arithmetic and analytic techniques.

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 K\"uhne. The main body of this paper consists of two parts: Part I for arithmetic estimates and Part II for analytic estimates.