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$p$-adic equidistribution and an application to $S$-units

Gerold Schefer

Abstract

We prove a Galois equidistribution result for torsion points in $\mathbb G_m^n$ in the $p$-adic setting for test functions of the form $\log |F|_p$ where $F$ is a nonzero polynomial with coefficients in the $p$-adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih's Conjecture is true for a class of divisors of $\mathbb G_m^n$.

$p$-adic equidistribution and an application to $S$-units

Abstract

We prove a Galois equidistribution result for torsion points in in the -adic setting for test functions of the form where is a nonzero polynomial with coefficients in the -adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih's Conjecture is true for a class of divisors of .

Paper Structure

This paper contains 17 sections, 36 theorems, 118 equations.

Table of Contents

  1. Abstract:
  2. Introduction
  3. Overview of the proofs
  4. Organisation of the article
  5. Acknowledgements
  6. Notation
  7. Univariate case
  8. Multivariate case
  9. $\Lambda_{\boldsymbol\zeta}$ and related definitions
  10. Some Galois theory
  11. Preparatory results
  12. Main theorem
  13. Ih's Conjecture
  14. $S$-integers, $S$-units and $p$-adic absolute values
  15. Archimedian part
  16. ...and 2 more sections

Key Result

Theorem 1

For each essentially atoral $P\in \overline{\mathbb Q}[X_1^{\pm1},\dots,X_n^{\pm 1}]\setminus \{0\}$ there exists $\kappa>0$ with the following property. Suppose $\boldsymbol\zeta\in\mathbb G_m^n(\mathbb C)$ has finite order and $\delta(\boldsymbol\zeta)$ is sufficiently large. Then $P(\sigma(\bolds as $\delta(\boldsymbol\zeta)\to\infty$, where the implicit constant depends only on $n$ and $P$.

Theorems & Definitions (85)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 75 more