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The Field of Definition for Dynamical Systems on $\mathbb P^N$
by Benjamin Hutz   Michelle Manes

Vol. 9 No. 4 (2014) P.585~P.602

ABSTRACT

Let $Hom^N_d$ be the set of morphisms $\phi: \mathbb{P}^{N} \to \mathbb{P}^{N}$ of degree $d$. For $f \in PGL_{N+1}$, let $\phi^f = f^{-1} \circ \phi \circ f$ be the conjugation action and let $M^N_d = Hom_d^N/PGL_{N+1}$ be the moduli space of degree $d$ morphisms of $P^N$. A field of definition for $\xi \in M_d^N$ is a field over which at least one representative $\phi \in \xi$ is defined. The field of moduli for $\xi$ is the fixed field of $G_\xi = \{\sigma \in Gal(\bar{K}/K) col \xi^{\sigma} = \xi\}$. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a field of definition for morphisms with trivial stabilizer group. Additionally, we use classical invariant theory to construct maps whose stabilizer group is non-trivial.

KEYWORDS
Arithmetic dynamics, field of definition, field of moduli, morphisms of $\mathbb P^N$.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 37P45, 11G99

MILESTONES