Abstract
Let K be a complete, algebraically closed non-Archimedean valued field, and let φ(z) ∈ K(z) have degree two. We describe the crucial set of φ in terms of the multipliers of φ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space M2(K) related to the reduction type of φ. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the classical non-Archimedean Julia set.
| Original language | English (US) |
|---|---|
| Article number | 11 |
| Journal | Research in Number Theory |
| Volume | 2 |
| Issue number | 1 |
| DOIs | |
| State | Published - Dec 1 2016 |
Funding
The research for this article was begun during an NSF-sponsored VIGRE research seminar on dynamics on the Berkovich line at the University of Georgia, led by the third author. Funding was provided by NSF Grant DMS-0738586. We would like to thank the other members of our research group for many helpful discussions, and would especially like to thank Jacob Hicks, Allan Lacy, Marko Milosevic, and Lori Watson for their insights and contributions to this work. We also thank the anonymous referees for suggestions concerning exposition.
Keywords
- Crucial set
- Moduli space
- Potential good reduction
- Quadratic map
- Stratification
ASJC Scopus subject areas
- Algebra and Number Theory