Module supersingulier et homologie des courbes modulaires

We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this Z-module to the homology of the modular curve X0(p). We give an interpretation of Gross formula for special values of L-functions of modular forms in this context. As an application, we obtain a formula for the sum of the squares of the optimal embeddings numbers of quadratic orders in a definite quaternion algebra.