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On relative fields of definition for log pairs, Vojta's height inequalities and asymptotic coordinate size dynamics

Nathan Grieve, Chatchai Noytaptim

TL;DR

This work develops a framework linking relative field of definition, log canonical pairs, and Vojta-type inequalities to study asymptotic coordinate-size dynamics in arithmetic dynamics on projective varieties. It introduces a generalized Vojta conjecture for log canonical pairs over finite field extensions and the notion of $ m F/K$ log resolutions, then derives a main dynamical result: for a surjective self-map $f$ on a variety with canonical singularities and a suitable divisor $D$, the coordinate-size contributions along $D$ become negligible relative to global height along generic orbits when $\alpha_f(x)>1$. The authors extend Matsuzawa (2023) to the setting of big line bundles and canonical singularities, and obtain Zariski-non-density results for forward-orbit integral points, connecting arithmetic dynamics with Diophantine inequalities. This provides a powerful approach to understanding asymptotic coordinate size dynamics and highlights the role of log resolutions and thresholds in dynamical Diophantine problems.

Abstract

We build on the perspective of the works \cite{Grieve:Noytaptim:fwd:orbits}, \cite{Matsuzawa:2023}, \cite{Grieve:qualitative:subspace}, \cite{Grieve:chow:approx}, \cite{Grieve:Divisorial:Instab:Vojta} (and others) and study the dynamical arithmetic complexity of rational points in projective varieties. Our main results make progress towards the attractive problem of asymptotic complexity of coordinate size dynamics in the sense formulated by Matsuzawa, in \cite[Question 1.1.2]{Matsuzawa:2023}, and building on earlier work of Silverman \cite{Silverman:1993}. A key tool to our approach here is a novel formulation of conjectural Vojta type inequalities for log canonical pairs and with respect to finite extensions of number fields. Among other features, these conjectured Diophantine arithmetic height inequalities raise the question of existence of log resolutions with respect to finite extensions of number fields which is another novel concept which we formulate in precise terms here and also which is of an independent interest.

On relative fields of definition for log pairs, Vojta's height inequalities and asymptotic coordinate size dynamics

TL;DR

This work develops a framework linking relative field of definition, log canonical pairs, and Vojta-type inequalities to study asymptotic coordinate-size dynamics in arithmetic dynamics on projective varieties. It introduces a generalized Vojta conjecture for log canonical pairs over finite field extensions and the notion of log resolutions, then derives a main dynamical result: for a surjective self-map on a variety with canonical singularities and a suitable divisor , the coordinate-size contributions along become negligible relative to global height along generic orbits when . The authors extend Matsuzawa (2023) to the setting of big line bundles and canonical singularities, and obtain Zariski-non-density results for forward-orbit integral points, connecting arithmetic dynamics with Diophantine inequalities. This provides a powerful approach to understanding asymptotic coordinate size dynamics and highlights the role of log resolutions and thresholds in dynamical Diophantine problems.

Abstract

We build on the perspective of the works \cite{Grieve:Noytaptim:fwd:orbits}, \cite{Matsuzawa:2023}, \cite{Grieve:qualitative:subspace}, \cite{Grieve:chow:approx}, \cite{Grieve:Divisorial:Instab:Vojta} (and others) and study the dynamical arithmetic complexity of rational points in projective varieties. Our main results make progress towards the attractive problem of asymptotic complexity of coordinate size dynamics in the sense formulated by Matsuzawa, in \cite[Question 1.1.2]{Matsuzawa:2023}, and building on earlier work of Silverman \cite{Silverman:1993}. A key tool to our approach here is a novel formulation of conjectural Vojta type inequalities for log canonical pairs and with respect to finite extensions of number fields. Among other features, these conjectured Diophantine arithmetic height inequalities raise the question of existence of log resolutions with respect to finite extensions of number fields which is another novel concept which we formulate in precise terms here and also which is of an independent interest.

Paper Structure

This paper contains 8 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Absolute values, local Weil and height functions
  3. Pairs, log resolutions and discrepancies
  4. Multiplier ideal sheaves and log canonical thresholds
  5. Arithmetic inequalities for log canonical pairs
  6. Preliminaries about arithmetic dynamical degrees
  7. Asymptotic dynamics of coordinate sizes along divisors
  8. Proof of Theorems \ref{['coord:sizes:main:thm:cor']} and \ref{['coord:sizes:ZD:main:thm:cor']}

Key Result

Theorem 1.5

Fix a finite set of places $S \subset M_{\mathbf{K}}$. Let $f \colon X \rightarrow X$ be a surjective morphism, defined over $\mathbf{K}$, for $X$ a projective variety over $\mathbf{K}$ and having canonical singularities. Let $D$ be a nonzero effective Cartier divisor on $X$ and defined over some fi

Theorems & Definitions (26)

  • Example 1.2
  • Conjecture 1.3
  • Conjecture 1.4: Compare for example with Bombieri:Gubler
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7: Grieve:Noytaptim:fwd:orbits
  • Theorem 1.8
  • Definition 3.1
  • Example 4.1
  • Definition 4.2
  • ...and 16 more