Critical Eisenstein series and Gross--Stark conjectures

In the first part of the talk, we explain how the study of the Coleman families passing through a critical p-stabilization of an Eisenstein series allows us to establish relations between different kinds of Euler systems. Next, we discuss some applications of this theory to the arithmetic of the adjoint of a weight one modular form. Beginning with the Gross--Stark conjecture for the adjoint p-adic L-function, which involves a unit and a p-unit, we discuss possible refinements in the spirit of the Harris-Venkatesh conjecture. Not surprisingly, a conceptual explanation for this phenomenon comes precisely for the degeneration of a triple product when one of the Coleman families specializes to a critical Eisenstein series. This circle of ideas contains the results of joint works (and works in progress) with David Loeffler, Alexandre Maksoud, Alice Pozzi and Victor Rotger.