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Some Applications of Dynamical Belyi Polynomials

Jacqueline Anderson, Michelle Manes, Bella Tobin

Abstract

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our proofs rely on conservative dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.

Some Applications of Dynamical Belyi Polynomials

Abstract

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our proofs rely on conservative dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.

Paper Structure

This paper contains 6 sections, 12 theorems, 45 equations, 1 table.

Table of Contents

  1. Background
  2. Dynamical systems definitions and notation
  3. Conservative dynamical Belyi polynomials
  4. Post-Critically Finite Polynomials and Bad Reduction
  5. Everywhere good reduction
  6. Heights of Conservative Polynomials

Key Result

Lemma \oldthetheorem

Let $f\in K[z]$ be a polynomial, and let $g$ be a polynomial conjugate of $f$ such that $g$ is monic and $g(0)=0$. Then $f$ has potential good reduction if and only if $g$ has good reduction.

Theorems & Definitions (30)

  • Definition \oldthetheorem
  • Definition \oldthetheorem: manesbelyi
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: MR1813109
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • proof : Proof of Theorem 1
  • ...and 20 more