p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou - I

Let p be an odd prime. Given an imaginary quadratic field K = Q(sqrt(−D_K) where p splits with D_K > 3, and a p-ordinary newform f ∈ Sk(Γ0(N)) such that N verifies the Heegner hypothesis relative to K, we prove a p-adic Gross–Zagier formula for the critical slope p-stabilization of f (assuming that it is non-θ-critical). In the particular case when f = fA is the newform of weight 2 associated to an elliptic curve A that has good ordi- nary reduction at p, this allows us to verify a conjecture of Perrin-Riou. The p-adic Gross–Zagier formula we prove has applications also towards the Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.First author draf