A Dynamical Analogue of the Criterion of N\'eron-Ogg-Shafarevich

TL;DR

This paper develops a dynamical analogue of the Néron-Ogg-Shafarevich criterion, linking unramified Galois representations to good reduction properties of rational maps and their critical loci, with applications to number field dynamics.

Contribution

It introduces a novel dynamical analogue of the Néron-Ogg-Shafarevich criterion using anabelian methods and explores its implications for arboreal Galois representations and ramification in number fields.

Findings

Unramified representations correspond to maps with good reduction at critical points.

Established criteria for primes with infinite ramification in arboreal representations.

Provided examples and applications to dynamical systems over number fields.

Abstract

We introduce an anabelian approach to the study of arboreal Galois representations and apply Tamagawa's anabelian version of the N\'eron-Ogg-Shafarevich criterion to produce a dynamical analogue of this criterion: unramified representations correspond to rational maps satisfying a strong form of good reduction in terms of their critical locus. Subsequently, we pursue a dynamical anlaogue of the N\'eron-Ogg-Shafarevich criterion in terms of the more (dynamically) traditional arboreal representations, which relates unramified arboreal representations to a certain separability condition on the dynamical system. Finally, we relate the our criteria: the anabelian criterion corresponds to the dynamical criterion as one varies the base point around the critical locus. Along the way we develop effective criteria to determine which primes are infinitely ramified in arboreal representations over number fields, as well as the asymptotic growth of that ramification; we conclude with examples and applications, especially to dynamical systems over number fields.