arXiv subjects

Phillip Williams

Publications and source records attributed to Phillip Williams.

4 recordsLinked to original sources

arXiv2016

Iteration and the Minimal Resultant

Let $K$ be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let $φ\in K(z)$ have degree $d\geq 2$. We characterize maps for which the minimal resultant of an iterate $φ^n$ is given by a simple formula in terms of $d$, $n$, and the minimal resultant of $φ$. We show that such maps are precisely those with reduction outside of an indeterminacy locus $I(d)$ and which also have semi-stable reduction for every iterate $φ^n$. We give two equivalent ways of describing such maps, one measure theoretic and the other in terms of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps. As an application, we are able to give an explicit formula for the minimal value of the diagonal Arakelov-Green's function of a map satisfying the conditions of the main theorem. We illustrate our results with some explicit calculations in the case of the Lattès maps.

math.DS↗

arXiv2014

Automorphism loci for the moduli space of rational maps

Let $k$ be an algebraically closed field of characteristic $0$ and $\mathcal{M}_d$ the moduli space of rational maps on $\mathbb{P}^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\subset \mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus $\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[ϕ]\in\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch-Popp-Oort characterization of singular points on the moduli scheme for curves. The results concerning these distinguished loci are used to compute the Picard and class groups of $\mathcal{M}_d, \mathcal{M}^s_d,$ and $\mathcal{M}^{ss}_d$.

math.DS↗

arXiv2012

Semi-stable Reduction Implies Minimality of the Resultant

For a dynamical system on n-dimensional projective space over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number field admits a globally minimal presentation.

math.DS↗

arXiv2012

Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable presentation coincides with the bad reduction (Theorem 3.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes map has unstable bad reduction (Theorem 3.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Theorem 2.1) to a natural dynamical analog to Theorem 1.1. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog to Theorem 1.1 may still be possible using the locus of critical bad reduction to define the conductor (Theorem 4.10, Theorem 4.13).

math.DS↗