Rational vs transcendental points on analytic Riemann surfaces

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 $M$.