The Saito-Kurokawa lifting and Darmon points

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Let E/ 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 { Aₜ(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 Stark-Heegner points) with the values of these Fourier coefficients and of their p-adic derivatives, evaluated at weight k=2.