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A Dynamical Analogue of Sen's Theorem

Ophelia Adams

TL;DR

The paper develops a dynamical analogue of Sen's theorem by linking the branch filtration of dynamical preimage extensions to the upper ramification filtration, up to a linear index change, under the regime of tamely ramification-stable branches. It proves that, for such branches, the infinite dynamical tower $K_ abla/K$ is arithmetically profinite and admits a precise description $K_n = K_ fty^{((V-1)n+1)}$ after a finite base change and index shift, with $V$ determined by limiting ramification data derived from the Newton polygon of the iterates. The work then identifies broad, effectively checkable conditions under which branches are tamely ramification-stable (notably when the degree is $p$ or the map is post-critically bounded with $p mid d$), and applies these results to prove partial answers to Berger's abelianization question and to questions about wild ramification in arboreal extensions, including showing infinite wild ramification in many prime-power cases. The paper also provides concrete algorithms for computing the limiting ramification data $(V,R,M,E,C)$ and demonstrates their use through an explicit sample calculation, highlighting both the practical potential and current limitations (notably the non-effectivity of the threshold $N$ in general). Overall, this work deepens the understanding of ramification in dynamical extensions and offers a computable framework for analyzing the intricate interaction between dynamics and arithmetic filtration.

Abstract

We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).

A Dynamical Analogue of Sen's Theorem

TL;DR

The paper develops a dynamical analogue of Sen's theorem by linking the branch filtration of dynamical preimage extensions to the upper ramification filtration, up to a linear index change, under the regime of tamely ramification-stable branches. It proves that, for such branches, the infinite dynamical tower is arithmetically profinite and admits a precise description after a finite base change and index shift, with determined by limiting ramification data derived from the Newton polygon of the iterates. The work then identifies broad, effectively checkable conditions under which branches are tamely ramification-stable (notably when the degree is or the map is post-critically bounded with ), and applies these results to prove partial answers to Berger's abelianization question and to questions about wild ramification in arboreal extensions, including showing infinite wild ramification in many prime-power cases. The paper also provides concrete algorithms for computing the limiting ramification data and demonstrates their use through an explicit sample calculation, highlighting both the practical potential and current limitations (notably the non-effectivity of the threshold in general). Overall, this work deepens the understanding of ramification in dynamical extensions and offers a computable framework for analyzing the intricate interaction between dynamics and arithmetic filtration.

Abstract

We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in -adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of -power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).

Paper Structure

This paper contains 13 sections, 19 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Prior work.
  3. Outline of the paper.
  4. Notation.
  5. Preliminaries
  6. Main Results
  7. Applications and Effectivity
  8. A question of Berger.
  9. A question about wild ramification in arboreal extensions.
  10. Effective results; calculating limiting ramification data.
  11. Calculating $V$, $R$, $M$, and $E$.
  12. Calculating $C$.
  13. Sample calculation.

Key Result

Theorem 1.1

Suppose our branch, associated to the polynomial $P(x)$ and base point $\alpha_0$, is tamely ramification-stable over $K$. Then $K_\infty/K$ is arithmetically profinite, and there is a constant $V$ such that for all $n$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 33 more