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 .