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Approximating rational points on surfaces

Brian Lehmann, David McKinnon, Matthew Satriano

Abstract

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations to $P$ must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.

Approximating rational points on surfaces

Abstract

Let be a smooth projective algebraic variety over a number field and in . In 2007, the second author conjectured that, in a precise sense, if rational points on are dense enough, then the best rational approximations to must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.
Paper Structure (8 sections, 14 theorems, 37 equations)

This paper contains 8 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Results
  4. Running the MMP
  5. Finding rational curves
  6. Finding rational curves: algebraically closed field
  7. Finding rational curves: number fields
  8. Surfaces

Key Result

Proposition 1.4

Let $X$ be a smooth, projective variety defined over a number field $k$, and let $P\in X(k)$. Then Vojta's Conjecture for $X$ implies that $P$ is essentially bounded.

Theorems & Definitions (37)

  • Definition 1.1
  • Conjecture 1.2: M2
  • Definition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 27 more