Derivatives of Beilinson–Flach classes, Gross–Stark formulas and a $p$-adic Harris–Venkatesh conjecture

Abstract

We propose an alternative approach to the study of exceptional zeros from the point of view of Euler systems. As a first application, we give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal{L}$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques, we discuss a new method exclusively using the properties of Beilinson–Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be seen as a counterpart to the earlier work of Loeffler, Venjakob, and Zerbes. In our second application, we illustrate how these techniques could lead to a better understanding of this setting by introducing a new motivic $p$-adic $L$-function whose special values encode information just about the unit of the adjoint (and not also the $p$-unit), in the spirit of the conjectures of Harris and Venkatesh. We further discuss conjectural connections with the arithmetic of triple products of Coleman families.