Quantitative Aspects of Arakelov Theory in Arithmetic Dynamics

Abstract

Let $K$ be a number field and $\vphi: \bb{P}^m \to \bb{P}^m$ be an endomorphism of degre $d \geq 2$ that is defined over $K$. Let $\h_{\vphi}: \bb{P}^m(\ovl{K}) \to \bb{R}{\geq 0}$ be the canonical height associated to $\vphi$. Given a sequence $(x_n) \in \bb{P}^m(\ovl{K})$, we say that it is generic if no hypersurface $Z$ contains infinitely many $x_n$'s. Yuan \cite{Yua08}, using Arakelov theory, proves that given a generic sequence of points $(x_n)$ with $\h{\vphi}(x_n) \to 0$ and a place $v \in M_K$, the Galois orbits of $x_n$ will equidistribute to the equilibrium measure $\mu_{\vphi,v}$. \par The aim of this thesis is to prove a quantitative version of Yuan's theorem for archimedean places. Given a smooth function $f: \bb{P}^m(\bb{C}) \to \bb{R}$ and an $\eps > 0$, we bound the degree of a hypersurface $Z(f,\eps)$ and a constant $\delta > 0$ such that $$\left|\frac{1}{|F_x|} \sum_{y \in F_x} f(y) - \int f d \mu_{\vphi,v} \right| \eps$$ holds for all $x \not \in Z(f,\eps)$ and $\h_{\vphi}(x) \delta$, where $F_x = \Gal(\ovl{K}/K) \cdot x$ is the Galois orbit of $x$. This upper bound on $\deg Z(f,\eps)$ tells us how generic $x$ has to be. \par There are two main new ingredients in the proof which follows Yuan's approach. The first is a quantitative form of the asymptotic expansion of the Bergman kernel, first established by Tian \cite{Tian90}, and the second is a construction of a ``dynamical" basis of polynomials due to Looper \cite{Loo24}. As an application, for $\bb{P}^2$ or smooth projective surfaces in general, we are able to deduce an exponential rate of convergence of $n$-periodic points $\Per_n$ to the equilibrium measure. A more arithmetic application is that we are able to deduce an exponential growth of the degree $[K(\Per_n):K]$ in terms of $n$, generalizing results due to Baker \cite{Bak06} in dimension one.