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 $p$-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.

Essentially final version, to appear in Transactions of AMS

References in corpus (1)

• Le système d'Euler de Kato en famille (I)

Cited by in corpus (5)

• $p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
• Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions
• On derivatives of Kato's Euler system for elliptic curves
• Iwasawa theory for Symmetric Square of non-$p$-ordinary eigenforms
• Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture