Rational versus transcendental points on analytic Riemann surfaces

Let (X, L) be a polarized variety over number field K. We suppose that L is an hermitian line bundle. M non compact Riemann Surface and $$U\subset M$$ relatively open set. $$\varphi :M\rightarrow X(\mathbf{C})$$ holomorphic map. For every positive real T, let $$A_U(T)$$ the cardinality of set $$z\in U$$ such (z)\in X(K)$$ $$h_L(\varphi (z))\le T$$ . After revisitation proof sub exponential bound for , obtained by Bombieri Pila, we show there are intervals reals T in these intervals, upper bounded polynomial T. then introduce subsets type S with respect $$ These which inequality similar to Liouville on algebraic points holds. that, if contains subset S, then, value As consequence, smooth leaf foliation curves defined K S(X) (full Lebesgue measure) verify some kind inequalities. In second part prove ^{-1}(S(X))\ne \emptyset only ^{-1}(S(X))$$ full measure M.