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Dynatomic Galois groups for a family of quadratic rational maps

David Krumm, Allan Lacy

Abstract

For every nonconstant rational function $φ\in\mathbb{Q}(x)$, the Galois groups of the dynatomic polynomials of $φ$ encode various properties of $φ$ that are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as $φ$ varies in a particular one-parameter family of maps, namely the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.

Dynatomic Galois groups for a family of quadratic rational maps

Abstract

For every nonconstant rational function , the Galois groups of the dynatomic polynomials of encode various properties of that are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as varies in a particular one-parameter family of maps, namely the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.
Paper Structure (20 sections, 23 theorems, 45 equations)

This paper contains 20 sections, 23 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Families of dynatomic groups
  3. Main results
  4. Methods
  5. Outline
  6. Computer code
  7. Preliminaries
  8. Specialization of Galois groups
  9. The moduli space $\mathcal{M}_d$
  10. The moduli space $\mathcal{M}_2$
  11. Dynatomic polynomials
  12. The third dynatomic group
  13. Maps with trivial automorphism group
  14. Maps with nontrivial automorphisms
  15. The fourth dynatomic group
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\phi\in\mathbb{Q}(x)$ be a rational function of degree $2$ having a $2$-periodic critical point, and suppose that $\operatorname{Aut}(\phi)$ is trivial. Then $\phi$ is linearly conjugate over $\mathbb{Q}$ to a map of the form Moreover, with notation as in Section trivial_section_G3, we have the following for $v\ne 3$: if $v$ is in the image of $\eta$, then $G_{3,\phi}\cong C$. Otherwise, $G_

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 33 more