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 -invariant of the adjoint of a weight one modular form in terms of units and -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 -adic -function whose special values encode information just about the unit of the adjoint (and not also the -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.
Cite this article
Óscar Rivero, Derivatives of Beilinson–Flach classes, Gross–Stark formulas and a -adic Harris–Venkatesh conjecture. Doc. Math. 28 (2023), no. 1, pp. 105–131DOI 10.4171/DM/905