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.

令 ( X ,  L ) 是数域K 上的极化变体。我们假设L是一个厄密线丛。设M是一个非紧黎曼曲面，\(U\subset M\)是一个相对紧的开集。令\(\varphi :M\rightarrow X(\mathbf{C})\)是一个全纯映射。对于每个正实数T，令\(A_U(T)\)是\(z\in U\)集合的基数，使得\(\varphi (z)\in X(K)\)和\ (h_L(\varphi (z))\le T\)。在重新审视 \(A_U(T)\)的次指数界的证明之后，由 Bombieri 和 Pila 获得，我们证明存在实数的区间，使得对于这些区间中的T，\(A_U(T)\)是T 中多项式的上限。然后我们引入关于\(\varphi \)的类型S 的子集。这些是M 的紧子集，其不等式类似于代数点上的刘维尔不等式。我们证明，如果M包含类型S的子集，那么对于T的每个值，数字\(A_U(T)\)都受T 中的多项式限制。因此，我们证明如果M 是在K 上定义的曲线中代数叶理的光滑叶，然后\(A_U(T)\)由T 中的多项式限定。令S ( X ) 是验证某种 Liouville 不等式的点的子集（Lebesgue 测度的完整点）。在第二部分，我们证明\(\varphi ^{-1}(S(X))\ne \emptyset \)当且仅当\(\varphi ^{-1}(S(X))\)是对M上的 Lebesgue 测度满。