Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire $X_0\left(N\right)$ comme un quotient du groupe ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$. En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de $N$ d’un relèvement de l’action des opérateurs de Hecke de $H_1(X_0\left(N\right),ptes,\mathbf{Z})$ sur ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$.

This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent $p$-adic modular symbols. Specifically, we give a constructive proof of acontrol theorem (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated $p$-adic $L$-functions in this case. In...