A Lehmer-type height lower bound for abelian surfaces over function fields

Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.

The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry

We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis

Post-Critically Finite Maps on $mathbb{P}^n$ for $nge2$ are Sparse

Let $f:{mathbb P}^nto{mathbb P}^n$ be a morphism of degree $dge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $kge1$ and $ellge0$ such that the critical locus $operatorname{Crit}_f$ satisfies $f^{k+ell}(operatorname{Crit}_f)subseteq{f^ell(operatorname{Crit}_f)}$. The smallest such $ell$ is called the tail-length. We prove that for $dge3$ and $nge2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $ell=0$, are not Zariski dense.

GIT Stability of Henon Maps

In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $operatorname{Rat}_d^N$ of degree $d$ rational maps $mathbb{P}^Ntomathbb{P}^N$ modulo the conjugation action of $operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $Nge3$ or $dge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $operatorname{Rat}_2^2$.

Moduli Spaces for Dynamical Systems with Portraits

A $textit{portrait}$ $mathcal{P}$ on $mathbb{P}^N$ is a pair of finite point sets $Ysubseteq{X}subsetmathbb{P}^N$, a map $Yto X$, and an assignment of weights to the points in $Y$. We construct a parameter space $operatorname{End}_d^N[mathcal{P}]$ whose points correspond to degree $d$ endomorphisms $f:mathbb{P}^Ntomathbb{P}^N$ such that $f:Yto{X}$ is as specified by a portrait $mathcal{P}$, and prove the existence of the GIT quotient moduli space $mathcal{M}_d^N[mathcal{P}]:=operatorname{End}_d^N//operatorname{SL}_{N+1}$ under the $operatorname{SL}_{N+1}$-action $(f,Y,X)^phi=bigl(phi^{-1}circ{f}circphi,phi^{-1}(Y),phi^{-1}(X)bigr)$ relative to an appropriately chosen line bundle. We also investigate the geometry of $mathcal{M}_d^N[mathcal{P}]$ and give two arithmetic applications.

A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on $mathbf{P}^N$

Let $f:mathbb{P}^Ntomathbb{P}^N$ be an endomorphism of degree $dge2$ defined over $overline{mathbb{Q}}$ or $overline{mathbb{Q}}_p$, and let $K$ be the field of moduli of $f$. We prove that there is a field of definition $L$ for $f$ whose degree $[L:K]$ is bounded solely in terms of $N$ and $d$.

Integrality properties of Bottcher coordinates for one-dimensional superattracting germs

Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $mge2$. The Bottcher coordinate of a power series $varphi(x)in x^m + x^{m+1}R[![x]!]$ is the unique power series $f_varphi(x)in x+x^2K[![x]!]$ satisfying $varphicirc f_varphi(x) = f_varphi(x^m)$. In this paper we study the integrality properties of the coefficients of $f_varphi(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) If $p$ is prime and $R=mathbb Z_p$ and $varphi(x)in x^p + px^{p+1}R[![x]!]$, then $f_varphi(x)in R[![x]!]$. (2) If $varphi(x)in x^m + mx^{m+1}R[![x]!]$, then $f_varphi(x)=xsum_{k=0}^infty a_kx^k/k!$ with all $a_kin R$. (3) In (2), if $m=p^2$, then $a_kequiv-1pmod{p}$ for all $k$ that are powers of $p$.

Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:mathbb{P}^1_Ktomathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^1_{R_S}tomathbb{P}^1_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^1_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^1_{R_S}$ is etale over $R_S$. We study triples $(f,Y,X)$ with $X=Ycup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $text{PGL}_2(R_S)$-equivalence classes of triples with $text{deg}(f)=d$ and $sum_{Pin Y}e_f(P)ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Yto X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$ when $Y=X$.

Degeneration of Dynamical Degrees in Families of Maps

The dynamical degree of a dominant rational map $f:mathbb{P}^Nrightarrowmathbb{P}^N$ is the quantity $delta(f):=lim(text{deg} f^n)^{1/n}$. We study the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make a conjecture and ask two questions concerning, respectively, the set of $t$ such that: (1) $delta(f_t)ledelta(f_T)-epsilon$; (2) $delta(f_t)<delta(f_T)$; (3) $delta(f_t)<delta(f_T)$ and $delta(g_t)<delta(g_T)$ for independent families of maps. We give a sufficient condition for our conjecture to hold and prove that it is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.