Rational vs transcendental points on analytic Riemann surfaces

TL;DR

This paper investigates the distribution of algebraic points on analytic Riemann surfaces, establishing polynomial bounds for their count within certain regions, and characterizes the preimages of Liouville-type sets under holomorphic maps.

Contribution

It introduces the concept of subsets of type S and proves polynomial bounds for algebraic points on Riemann surfaces containing such subsets, extending previous subexponential bounds.

Findings

Polynomial bounds for algebraic points on Riemann surfaces

Existence of subsets of type S with Liouville inequality properties

Characterization of preimages of Liouville sets under holomorphic maps

Abstract

Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $\varphi:M\to X({\Bbb C})$ be a holomorphic map. For every positive real number $T$, let $A_U(T)$ be the cardinality of the set of $z\in U$ such that $\varphi (z)\in X(K)$ and $h_L(\varphi(z))\leq T$. After a revisitation of the proof of the sub exponential bound for $A_U(T)$, obtained by Bombieri and Pila , we show that there are intervals of $T$'s as big as we want for which $A_U(T)$ is upper bounded by a polynomial in $T$. We then introduce subsets of type $S$ with respect of $\varphi$. These are compact subsets of $M$ for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if $M$ contains a subset of type $S$, then, {\it for every value of $T$} the number $A_U(T)$ is bounded by a polynomial in $T$. As a consequence, we show that if $M$ is a smooth leaf of a foliation in curves then $A_U(T)$ is bounded by a polynomial in $T$. Let $S(X)$ be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that $\varphi^{-1}(S(X))\neq\emptyset$ if and only if $\varphi^{-1}(S(X))$ is full for the Lebesgue measure on $M$.