The Saito–Kurokawa lifting and Darmon points

International audienceLet E/ℚ be an elliptic curve of conductor Np with p∤N and let f be its associated newform of weight 2. Denote by f∞ the p-adic Hida family passing though f, and by F∞ its Λ-adic Saito–Kurokawa lift. The p-adic family F∞ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients {A˜T(k)}T indexed by positive definite symmetric half-integral matrices T of size 2×2. We relate explicitly certain global points on E (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their p-adic derivatives, evaluated at weight k=2