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

Abstract

Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f\in S_k({\Gamma }_0(N))$ 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-$\theta$-critical). In the particular case when $f=f_A$ is the newform of weight $2$ associated to an elliptic curve $A$ that has good ordinary 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.

arXiv comment40 pages. Most significant updates are in Section 4

References in corpus (3)

• On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou
• P-adic heights and p-adic Hodge theory
• Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms

Cited by in corpus (8)

• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
• The universal $p$-adic Gross-Zagier formula
• On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes
• Interpolation of Generalized Heegner Cycles in Coleman Families
• Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions
• On derivatives of Kato's Euler system for elliptic curves
• On derivatives of Kato's Euler system and the Mazur-Tate Conjecture