Explicit cocycle of the Dedekind-Rademacher cohomology class and the Darmon-Dasgupta measures

Abstract

The work of Darmon, Pozzi, and Vonk [3] has recently shown that the RM-values of the Dedekind-Rademacher cocycle $J_{DR}$ are Gross-Stark units up to controlled torsion. The authors of [3] remarked that the measure-valued cohomology class, which we denote ${\mu }_{DR}^p$, underlying $J_{DR}$ is the level 1 incarnation of earlier constructions in [1]. In this paper, we make this relationship explicit by computing a concrete cocycle representative of an adelic incarnation ${\mu }_{DR}$ by tracing the construction of the cohomology class and comparing periods of weight 2 Eisenstein series. While maintaining a global perspective in our computations, we configure the appropriate method of smoothing cocycles which exactly yields the p-adic measures of [1] when applied to ${\mu }_{DR}$. These methods will also explain the optional degree zero condition imposed in [1] which was remarked upon in [6] and [7].