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On the set of stable primes for postcritically infinite maps over number fields

Joachim König

TL;DR

The paper studies when iterates of a polynomial or rational function over a number field remain irreducible mod almost all primes, i.e., the stability of $f$ and its fibers. It proves unconditional density-zero results for broad classes of postcritically infinite maps by combining group-theoretic control of imprimitive Galois actions (via iterated wreath products) with ramification data from the Specialization Inertia Theorem and prime divisors in dynamical sequences, translating cycle-structure constraints into density statements via Chebotarev. Notably, it shows that for many (in particular, most odd-degree) polynomials the set of stable primes has density zero, and it establishes a dichotomy for unicritical PCF maps: stability has positive density outside a thin set for PCF unicritical cases, while non-PCF or wandering-point hypotheses force density zero. These results advance the understanding of Arboreal representations and support the conjecture that large sets of stable primes imply highly constrained, PCF-like shapes. The techniques provide unconditional evidence toward the broader expectation that nontrivial stability occurs only in very special dynamical configurations.

Abstract

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of $f$ are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to $100\%$ of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a "large" set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.

On the set of stable primes for postcritically infinite maps over number fields

TL;DR

The paper studies when iterates of a polynomial or rational function over a number field remain irreducible mod almost all primes, i.e., the stability of and its fibers. It proves unconditional density-zero results for broad classes of postcritically infinite maps by combining group-theoretic control of imprimitive Galois actions (via iterated wreath products) with ramification data from the Specialization Inertia Theorem and prime divisors in dynamical sequences, translating cycle-structure constraints into density statements via Chebotarev. Notably, it shows that for many (in particular, most odd-degree) polynomials the set of stable primes has density zero, and it establishes a dichotomy for unicritical PCF maps: stability has positive density outside a thin set for PCF unicritical cases, while non-PCF or wandering-point hypotheses force density zero. These results advance the understanding of Arboreal representations and support the conjecture that large sets of stable primes imply highly constrained, PCF-like shapes. The techniques provide unconditional evidence toward the broader expectation that nontrivial stability occurs only in very special dynamical configurations.

Abstract

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a "large" set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.
Paper Structure (8 sections, 16 theorems, 4 equations)

This paper contains 8 sections, 16 theorems, 4 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Prerequisites
  3. Some basics of imprimitive permutation groups
  4. Previous results on mod-$p$ stability
  5. Ramification in specializations and prime divisors of dynamical sequences
  6. Proofs of the main results
  7. A result for "large" arboreal representations
  8. Some remarks on low degree polynomials

Key Result

Theorem 1.1

Let $f\in K(X)$ be a postcritically infinite rational function fulfilling the following: There exists some critical point $\alpha$ of $f$ which is a wandering point and whose multiplicity under $f$ is divisible by some prime number coprime to $\deg(f)$. Then the set of stable primes of $f$ is of den

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • ...and 26 more