Module supersingulier et homologie des courbes modulaires Original Research Article

We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this image-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.