On the exceptional zeros of 𝑝-non-ordinary 𝑝-adic 𝐿-functions and a conjecture of Perrin-Riou

Abstract

Our goal in this article is to prove a form of p p -adic Birch and Swinnerton-Dyer formula for the second derivative of the p p -adic L L -function associated to a newform f f which is non-crystalline semistable at p p at its central critical point, by expressing this quantity in terms of a p 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 f . This, among other things, leads us to a proof of an appropriate version of Perrin-Riou’s conjecture in this set up.