Minimal resultant locus and its moduli theoretic characterization in non-archimedean dynamics

Abstract

Let φ be a rational function (of degree more than 1) on the projective line ℙ¹ over an algebraically closed and complete non-trivial and non-archimedean valued field 𝘒, which is an endomorphism of ℙ¹. The degree of the reduction of φ modulo the maximal ideal in the ring of 𝘒-integers is less than or equal to that of φ, and we say φ has a good reduction if the equality holds. A conjugacy of φ under some projective transformation of ℙ¹ can have a good reduction even if so does not φ. The minimal resultant locus for φ is a dynamical equivariant which measures how far φ is from having a good reduction, up to conjugations of it under projective transformations. In this talk, after reviewing the foundational moduli theoretic works by Rumely, Szpiro-Tepper-Williams, Silverman, ... on the minimal resultant locus (to characterize the minimum resultant locus as the potential GIT-semistable locus), we introduce the hyperbolic resultant function for φ on the Berkovich projective line over 𝘒 and the intrinsic depths of the intrinsic reduction of φ at each point of the Berkovich projective line. The main result is the moduli theoretic characterization of the minimal resultant locus of φ using the Berkovich hyperbolic geometry.