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Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields

Christoph Pütz

TL;DR

The paper classifies Laurent series over non-Archimedean local fields that map infinitely many roots of unity to roots of unity, showing such series are forced to be 'special' (simple monomial form in characteristic 0 or finite-field-like in positive characteristic). It develops a ramified/unramified analytic framework and leverages the auxiliary function $f(X^q)-f(X)^q$ to bound torsion on the graph, yielding effective, computable bounds tied to zeros of unit-valued functions. In positive characteristic, the bound is governed by the count of zeros of $f(X)^q-f(X^q)$; in characteristic 0, ramification and a computable constant $c_f$ appear, producing explicit finite bounds and applications to Manin–Mumford-type problems in $\mathbb{G}_m^2$. The results extend Schmidt’s ideas to Laurent series on unit spheres and provide practical bounds for torsion configurations relevant to arithmetic dynamics and algebraic geometry over $p$-adic fields.

Abstract

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series $f$ over a field of positive characteristic with residue field $\mathbb{F}_q$, we prove effective bounds for the number of possible roots of unity in terms of the number of zeroes of the auxilliary function $f(x^q)-f(x)^q$ on the unit circle. In characteristic $0$ our bound is still effective but also depends on the ramification degree of the base field over $\mathbb{Q}_p$ as well as the size of the coefficients of $f$. This has applications to the Manin-Mumford conjecture in $\mathbb{G}_m^2$. In characteristic $0$, this work builds upon a pigeon-hole based method by Schmidt.

Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields

TL;DR

The paper classifies Laurent series over non-Archimedean local fields that map infinitely many roots of unity to roots of unity, showing such series are forced to be 'special' (simple monomial form in characteristic 0 or finite-field-like in positive characteristic). It develops a ramified/unramified analytic framework and leverages the auxiliary function to bound torsion on the graph, yielding effective, computable bounds tied to zeros of unit-valued functions. In positive characteristic, the bound is governed by the count of zeros of ; in characteristic 0, ramification and a computable constant appear, producing explicit finite bounds and applications to Manin–Mumford-type problems in . The results extend Schmidt’s ideas to Laurent series on unit spheres and provide practical bounds for torsion configurations relevant to arithmetic dynamics and algebraic geometry over -adic fields.

Abstract

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series over a field of positive characteristic with residue field , we prove effective bounds for the number of possible roots of unity in terms of the number of zeroes of the auxilliary function on the unit circle. In characteristic our bound is still effective but also depends on the ramification degree of the base field over as well as the size of the coefficients of . This has applications to the Manin-Mumford conjecture in . In characteristic , this work builds upon a pigeon-hole based method by Schmidt.

Paper Structure

This paper contains 9 sections, 23 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-Archimedean Analysis
  4. $p$-adic number fields
  5. Theorem A
  6. The ramified part
  7. The unramified part
  8. Effective bounds
  9. Acknowledgments

Key Result

Theorem 1.1

[thm]thm:ist Let $K$ be a number field and $P\in K[X,Y]$ be a polynomial such that is infinite. Then there are coprime non-negative integers $r,s$ such that $X^rY^s-1$ or $X^r-Y^s$ are not coprime to $P(X,Y)$.

Theorems & Definitions (42)

  • Theorem 1.1: Ihara, Serre and Tate, lang_65
  • Corollary 1.2
  • Theorem A
  • Definition 2.1: Analytic Laurent series, cherry
  • Definition 2.2: Special Laurent series
  • Theorem A
  • Theorem 2.3: Appendix A, Lemma 6 in bosch
  • Definition 2.4: The absolute values $|\cdot|_r$
  • Definition 2.5: $K(f,r)$ and $k(f,r)$
  • Lemma 2.6: cherry
  • ...and 32 more