The exceptional zero phenomenon for elliptic units

TL;DR

This paper investigates the exceptional zero phenomenon for elliptic units in imaginary quadratic fields, deriving explicit formulas linking derived units to $p$-adic $L$-functions and $ ext{L}$-invariants, thus connecting various approaches in number theory.

Contribution

It provides an explicit formula relating derived elliptic units to Katz's $p$-adic $L$-function and interprets this in terms of an $ ext{L}$-invariant, extending understanding of the phenomenon.

Findings

Explicit formula linking derived elliptic units to $p$-adic $L$-values

Interpretation of the phenomenon via $ ext{L}$-invariants

Connections to Heegner points and Beilinson--Flach elements

Abstract

The exceptional zero phenomenon has been widely studied in the realm of $p$-adic $L$-functions, where the starting point lies in the foundational work of Mazur, Tate and Teitelbaum. This phenomenon also appears in the study of Euler systems, which comes as no surprise given the interaction between these two settings. When this occurs, one is led to study higher order derivatives of the Euler system in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a {\it derived} elliptic unit either to special values of Katz's two variable $p$-adic $L$-function or to its derivatives. Further, we interpret this fact in terms of an $\mathcal L$-invariant, and relate this result to other approaches to the exceptional zero phenomenon concerning Heegner points and Beilinson--Flach elements.