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

 I will report joint work in progress with R. Pollack and S. Sasaki, where we prove a critical slope p-adic Gross-Zagier formula for critical slope p-adic L-functions. Besides the strategy for our proof, I will illustrate a number of applications. The first of these is the proof of a conjecture of Perrin-Riou (that predicts an explicit relation between Beilinson-Kato elements and Heegner points). The second is towards a Birch and Swinnerton-Dyer formula when the analytic rank is one (yielding a simplified proof of the recent results of Jetchev-Skinner-Wan in this context).