Exceptional zero formulae and a conjecture of Perrin-Riou

TL;DR

This paper proves new cases of conjectures relating $p$-adic $L$-functions, Heegner points, and elliptic curves, establishing a $p$-adic Gross-Zagier formula and confirming parts of the exceptional zero conjecture.

Contribution

It establishes an analogue of Perrin-Riou's conjecture, relates $p$-adic Beilinson-Kato elements to Heegner points, and proves a $p$-adic Gross-Zagier formula for elliptic curves with split multiplicative reduction.

Findings

Proved an analogue of Perrin-Riou's conjecture for elliptic curves.

Established a $p$-adic Gross-Zagier formula relating derivatives of $p$-adic $L$-functions to Heegner points.

Confirmed a large part of the rank-one case of the Mazur-Tate-Teitelbaum exceptional zero conjecture.

Abstract

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\infty}$. We prove a $p$-adic Gross$-$Zagier formula, expressing the quadratic term of the Taylor expansion of $L_{p}(f_{\infty},k,s)$ at $(k,s)=(2,1)$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $A(\mathbb{Q})$.