Rational versus transcendental points on analytic Riemann surfaces

Abstract

Let (X, L) be a polarized variety over a number field K. 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 $\varphi :M\rightarrow X(\mathbf{C})$ 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 $h_L(\varphi (z))\le T$. 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 the reals such that for T in these intervals, $A_U(T)$ 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, 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 an algebraic foliation in curves defined over K 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 $\varphi ^{-1}(S(X))\ne \emptyset$ if and only if $\varphi ^{-1}(S(X))$ is full for the Lebesgue measure on M.