Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitativity on the number of rational points in the Mordell conjecture

Jiawei Yu, Xinyi Yuan, Shengxuan Zhou

TL;DR

This work delivers an explicit quantitative bound on the number of rational points on curves of genus $g\ge2$ over number fields, tightly linking Mordell’s finiteness to explicit constants. The authors develop a two-pronged approach: arithmetic estimates anchored in adelic and Néron–Tate height theory, and analytic estimates that control Green functions, invariant quantities, and hyperbolic geometry via the $\varphi$-invariant and FE invariant. Central contributions include an explicit bound $\#((C(K)-\alpha)\cap \Lambda)\le 10^{13}g^{8}\cdot \min\{1+\frac{5}{4\sqrt g},1+\frac{3\log g}{g}\}^{\mathrm{rk}(\Lambda)}$, plus a quantitative Manin–Mumford statement and a quantitative Bogomolov-type framework, with thorough function-field analogues. The methods blend Vojta’s inequality, sphere-packing refinements, and precise control of admissible adelic line bundles, yielding practical, explicit constants that sharpen our understanding of uniform Mordell-type bounds and their arithmetic-geometric underpinnings. The results have potential implications for hyperelliptic families, average-point counts, and further explicit arithmetic geometry in both number field and function field settings.

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.

Quantitativity on the number of rational points in the Mordell conjecture

TL;DR

This work delivers an explicit quantitative bound on the number of rational points on curves of genus over number fields, tightly linking Mordell’s finiteness to explicit constants. The authors develop a two-pronged approach: arithmetic estimates anchored in adelic and Néron–Tate height theory, and analytic estimates that control Green functions, invariant quantities, and hyperbolic geometry via the -invariant and FE invariant. Central contributions include an explicit bound , plus a quantitative Manin–Mumford statement and a quantitative Bogomolov-type framework, with thorough function-field analogues. The methods blend Vojta’s inequality, sphere-packing refinements, and precise control of admissible adelic line bundles, yielding practical, explicit constants that sharpen our understanding of uniform Mordell-type bounds and their arithmetic-geometric underpinnings. The results have potential implications for hyperelliptic families, average-point counts, and further explicit arithmetic geometry in both number field and function field settings.

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.
Paper Structure (55 sections, 124 theorems, 1092 equations)

This paper contains 55 sections, 124 theorems, 1092 equations.

Table of Contents

  1. Introduction
  2. Main theorem: quantitative Mordell conjecture
  3. Some consequences
  4. Quantitative Vojta inequality
  5. Quantitative Bogomolov conjecture
  6. Sphere packing
  7. Systoles of hyperbolic surfaces
  8. Estimate of the Lg-invariant
  9. Estimate of the Faltings--Elkies invariant
  10. Notation and terminology
  11. Arithmetic Estimates
  12. Preliminaries on arithmetic intersection theory
  13. Néron--Tate heights
  14. Intersection theory of hermitian line bundles
  15. Metrized line bundles on Berkovich analytic spaces
  16. ...and 40 more sections

Key Result

Theorem 1.1

Let $K$ be a field of characteristic $0$, and let $C$ be a curve of genus $g \geq 2$ over $K$. Denote by $J$ the Jacobian variety of $C$. Let $\Lambda \subset J(K)$ be a subgroup of finite rank. Fix a point $P_0 \in C(K)$, and view $C(K)-P_0$ as a subset of $J(C)(K)$ via the Abel--Jacobi embedding. where $c_1(g)$ and $c_2(g)$ are positive constants depending only on the genus $g$.

Theorems & Definitions (224)

  • Theorem 1.1: Vojta Vojta_Ar, Dimitrov--Gao--Habegger DGH, Kühne Kuhne
  • Theorem 1.2: Theorem \ref{['mordell_lang']}
  • Theorem 1.3
  • Theorem 1.4: Theorem \ref{['manin_mumford']}
  • Theorem 1.5: Theorem \ref{['geometric_mordell_lang']}
  • Theorem 1.6
  • Theorem 1.7: Theorem \ref{['hyperelliptic']}
  • Theorem 1.8: Theorem \ref{['average']}
  • Theorem 1.9: Theorem \ref{['vojta']}
  • Theorem 1.10: Theorem \ref{['mumford']}
  • ...and 214 more