Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

TL;DR

This paper extends the understanding of anticyclotomic p-adic L-functions for elliptic curves to the ramified prime case, establishing an exceptional zero formula relating derivatives of L-functions to elliptic curve points.

Contribution

It develops a systematic study of anticyclotomic p-adic L-functions when p is ramified in K, generalizing previous inert case results and establishing a new exceptional zero formula.

Findings

Derived an exceptional zero formula for ramified primes p in K.

Connected the derivative of p-adic L-functions to formal group logarithms of elliptic curve points.

Extended the theory to cases with multiplicative reduction at p.

Abstract

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional zeroes phenomenon investigated by Bertolini-Darmon in the setting when $p$ is inert in $K$. More precisely, we consider situations in which the $p$-adic $L$-function $L_p(E/K)$ of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ does not vanish identically but, by sign reasons, has a zero at certain characters $\chi$ of the Hilbert class field of $K$. In this case we show that the value at $\chi$ of the first derivative of $L_p(E/K)$ is equal to the formal group logarithm of the specialization at $p$ of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini-Darmon, available when $p$ is inert in $K$ and $\chi$ is the trivial character.