Rational versus transcendental points on analytic Riemann surfaces

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⊂ M be a relatively compact open set. Let φ :M→ X(𝐂) be a holomorphic map. For every positive real number T , let A_U(T) be the cardinality of the set of z∈ U such that φ (z)∈ X(K) and h_L(φ (z))≤ 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 φ . 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 φ ^-1(S(X))∅ if and only if φ ^-1(S(X)) is full for the Lebesgue measure on M .