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On Vojta's proof of the Mordell conjecture

Xinyi Yuan

TL;DR

The paper reframes Vojta's original Mordell conjecture proof within Arakelov geometry, replacing Gillet–Soulé's arithmetic Riemann–Roch with Yuan's arithmetic Siu inequality. It constructs a small section of a high multiple of a Vojta line bundle using arithmetic Siu to obtain a height lower bound, and then derives a lower bound for the point heights via a careful analysis of vanishing indices. The index is controlled using a classical Dyson-type lemma (via three reductions and a blow-up argument), completing the chain that yields Vojta's inequality and the finiteness of $C(K)$. The approach preserves the Arakelov-geometric spirit while avoiding the arithmetic Riemann–Roch step, offering a streamlined route to Faltings' theorem and related uniform bounds in Diophantine geometry.

Abstract

This paper re-organizes Vojta's proof of the Mordell conjecture (i.e. Faltings' theorem) in terms of Arakelov geometry. A new ingredient is to replace an application of Gillet--Soule's arithmetic Riemannn--Roch theorem by that of Yuan's arithmetic Siu inequality.

On Vojta's proof of the Mordell conjecture

TL;DR

The paper reframes Vojta's original Mordell conjecture proof within Arakelov geometry, replacing Gillet–Soulé's arithmetic Riemann–Roch with Yuan's arithmetic Siu inequality. It constructs a small section of a high multiple of a Vojta line bundle using arithmetic Siu to obtain a height lower bound, and then derives a lower bound for the point heights via a careful analysis of vanishing indices. The index is controlled using a classical Dyson-type lemma (via three reductions and a blow-up argument), completing the chain that yields Vojta's inequality and the finiteness of . The approach preserves the Arakelov-geometric spirit while avoiding the arithmetic Riemann–Roch step, offering a streamlined route to Faltings' theorem and related uniform bounds in Diophantine geometry.

Abstract

This paper re-organizes Vojta's proof of the Mordell conjecture (i.e. Faltings' theorem) in terms of Arakelov geometry. A new ingredient is to replace an application of Gillet--Soule's arithmetic Riemannn--Roch theorem by that of Yuan's arithmetic Siu inequality.

Paper Structure

This paper contains 22 sections, 18 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Vojta's inequality
  3. Vojta's inequality
  4. Mumford's equality
  5. A precise statement
  6. The Arakelov-geometric setting
  7. Basic Arakelov geometry
  8. Arithmetic models
  9. Plan of proof
  10. Step 1: existence of small section
  11. Hilbert--Samuel type of estimates
  12. Estimate of the arithmetic Euler characteristic
  13. Step 2: lower bound of the height
  14. The hermitian line bundle by restriction
  15. The section of the line bundle
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $C$ be a smooth, projective, and geometrically connected curve of genus $g>1$ over a number field $K$. Then $C(K)$ is finite.

Theorems & Definitions (30)

  • Theorem 1.1: Mordell conjecture
  • Theorem 2.1: Vojta's inequality
  • Theorem 2.2: Mumford's equality
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • Theorem 3.1: small section
  • ...and 20 more