Related papers: A Uniform Field-of-Definition/Field-of-Moduli Boun…

Let Hom^N_d be the set of morphisms of degree d from P^N to itself. For f an element of PGL_{N+1}, let phi^f represent the conjugation action f^{-1} phi f. Let M^N_d = Hom_d^N/PGL_{N+1} be the moduli space of degree d morphisms of P^N. A…

We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on $\mathbb{P}^{n}$. We prove that, for fixed $n$, there exists a constant $C_{n}$ such that every dynamical system…

J. Silverman proved that a dynamical system on $\mathbb{P}^{1}$ descends to the field of moduli if it is polynomial or it has even degree, but for non-polynomial ones of odd degree the picture is less clear. We give a complete…

Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial…

Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq\overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of…

As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose…

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that…

Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms…

For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the M\"obius transformations that commute with $f$, and these form a finite subgroup of…

We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.

It is a classical fact going back to F. Klein that an elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$,…

We prove that, over any field, the dimension of the indeterminacy locus of a rational transformation $f$ of $P^n$ which is defined by monomials of the same degree $d$ with no common factors is at least $(n-2)/2$, provided that the degree of…

It is known that sometimes a Belyi pair is not defined over its field of moduli. Instead, it is defined over a finite degree extension of its field of moduli, called a field of definition. We show that given a number $m$ there exists a…

If $X$ is a variety with an additional structure $\xi$, such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There…

In this paper we investigate the cohomological obstruction for the field of moduli of a $G$-cover to be a field of definition, in the case of local fields and covers with tame admissible reduction. This applies in particular to $p$-adic…

A $\textit{portrait}$ $\mathcal{P}$ on $\mathbb{P}^N$ is a pair of finite point sets $Y\subseteq{X}\subset\mathbb{P}^N$, a map $Y\to X$, and an assignment of weights to the points in $Y$. We construct a parameter space…

In this paper, we study the relation between two dynamical systems (V,f) and (V,g) with f. g = g . f. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Pre(f) has only finitely…

We prove a dynamical analogue of the Shafarevich conjecture for morphisms $f:\mathbb{P}_K^N\to\mathbb{P}_K^N$ of degree $d\ge2$, defined over a number field $K$. This extends previous work of Silverman and others in the case $N=1$.

Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this…

We study the field of moduli of singular abelian and K3 surfaces. We discuss both the field of moduli over the CM field and over $\Q$. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an…