ON THE EXCEPTIONAL ZEROS OF p-NON-ORDINARY p-ADIC L-FUNCTIONS AND A CONJECTURE OF PERRIN-RIOU

Our goal in this article is to prove a form of p-adic Birch and Swinnerton-Dyer formula for the second derivative of the p-adic L-function associated to a newform f which is non-crystalline semistable at p at its central critical point, by expressing this quantity in terms of a p-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through f. This, among other things, leads us to a proof of an appropriate version of Perrin-Riou's conjecture in this set up.