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A New Diophantine Approximation Inequality on Surfaces and Its Applications

Keping Huang, Aaron Levin, Zheng Xiao

Abstract

We prove a Diophantine approximation inequality for closed subschemes on surfaces which can be viewed as a joint generalization of recent inequalities of Ru-Vojta and Heier-Levin in this context. As applications, we study various Diophantine problems on affine surfaces given as the complement of three numerically parallel ample projective curves: inequalities involving greatest common divisors, degeneracy of integral points, and related Diophantine equations including families of S-unit equations. We state analogous results in the complex analytic setting, where our main result is an inequality of Second Main Theorem type for surfaces, with applications to the study and value distribution theory of holomorphic curves in surfaces.

A New Diophantine Approximation Inequality on Surfaces and Its Applications

Abstract

We prove a Diophantine approximation inequality for closed subschemes on surfaces which can be viewed as a joint generalization of recent inequalities of Ru-Vojta and Heier-Levin in this context. As applications, we study various Diophantine problems on affine surfaces given as the complement of three numerically parallel ample projective curves: inequalities involving greatest common divisors, degeneracy of integral points, and related Diophantine equations including families of S-unit equations. We state analogous results in the complex analytic setting, where our main result is an inequality of Second Main Theorem type for surfaces, with applications to the study and value distribution theory of holomorphic curves in surfaces.

Paper Structure

This paper contains 11 sections, 35 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Heights Associated to Closed Subschemes
  4. Multiplicities, Regular Sequences, and the Filtration Lemma
  5. Beta constant estimates
  6. Main Result and Some Consequences
  7. Applications
  8. Greatest Common Divisors on Surfaces
  9. On the Diophantine Equation $f(a^m, y) = b^n$
  10. Integral Points on the Complement of Three Numerically Parallel Curves Passing Through a Point
  11. Families of Unit Equations

Key Result

Theorem 1.1

Let $S$ be a finite set of places of a number field $K$. For each $v\in S$, let $H_{0,v},\ldots,H_{n,v}\subset \mathbb{P}^n$ be hyperplanes over $K$ in general position. Let $\varepsilon>0$. Then there exists a finite union of hyperplanes $Z\subset \mathbb{P}^n$ such that for all points $P\in \mathb

Theorems & Definitions (77)

  • Theorem 1.1: Schmidt Subspace Theorem
  • Theorem 1.2: Ru-Vojta RV20, Ru-Wang RW22, Vojta Voj23
  • Theorem 1.3: Heier-Levin HL21
  • Theorem 1.4: Main Theorem
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1
  • Remark 2.2
  • ...and 67 more