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An Explicit Uniform Mordell Conjecture over Function Fields of Characteristic Zero

Jiawei Yu

TL;DR

The paper proves an explicit uniform Mordell-type bound for non-isotrivial curves of genus $g>1$ over function fields of characteristic zero, bounding the number of rational points in terms of the genus and the rank of the Jacobian's rational points. The method blends Vojta's Diophantine approach with Zhang's adelic line bundles and a quantitative Bogomolov result of Looper–Silverman–Wilms, together with a careful height comparison between the admissible and canonical heights via Green functions on reduction graphs. Key ingredients include Siu's inequality for positivity of big line bundles, Dyson-type index estimates, and a lattice-ball counting argument that yields an explicit bound with constants depending only on $g$ and the Lang-Néron rank $\rho$. The results give an effective, explicit uniform finiteness statement in the function-field setting, advancing our understanding of height-based uniformity in arithmetic geometry.

Abstract

We give an explicit uniform result on the Mordell conjecture for non-isotrivial curves over function field of characteristic 0. The proof is based on Vojta's method, and make use of Zhang's admissible adelic line bundles and a quantitative proof of the Bogomolov conjecture by Looper-Silverman-Wilms.

An Explicit Uniform Mordell Conjecture over Function Fields of Characteristic Zero

TL;DR

The paper proves an explicit uniform Mordell-type bound for non-isotrivial curves of genus over function fields of characteristic zero, bounding the number of rational points in terms of the genus and the rank of the Jacobian's rational points. The method blends Vojta's Diophantine approach with Zhang's adelic line bundles and a quantitative Bogomolov result of Looper–Silverman–Wilms, together with a careful height comparison between the admissible and canonical heights via Green functions on reduction graphs. Key ingredients include Siu's inequality for positivity of big line bundles, Dyson-type index estimates, and a lattice-ball counting argument that yields an explicit bound with constants depending only on and the Lang-Néron rank . The results give an effective, explicit uniform finiteness statement in the function-field setting, advancing our understanding of height-based uniformity in arithmetic geometry.

Abstract

We give an explicit uniform result on the Mordell conjecture for non-isotrivial curves over function field of characteristic 0. The proof is based on Vojta's method, and make use of Zhang's admissible adelic line bundles and a quantitative proof of the Bogomolov conjecture by Looper-Silverman-Wilms.
Paper Structure (4 sections, 8 theorems, 95 equations)

This paper contains 4 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Difference between heights
  3. Vojta's inequality
  4. Proof of the main theorem

Key Result

Theorem 1.1

For any integer $g>1$, there is a constant $c(g)$ with the following property. Let $K$ be a field of characteristic $0$, $C/K$ a geometrically connected smooth projective curve of genus $g$, $J$ the Jacobian variety of $C/K$, and $P_0\in C(K)$ a rational point. Then for any subgroup $\Gamma\subseteq

Theorems & Definitions (14)

  • Theorem 1.1: Dimitrov-Gao-Habegger+Kühne
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['ineq']}.
  • ...and 4 more