The values of the Dedekind-Rademacher cocycle at real multiplication points

The values of the so-called {\em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight one in the anti-parallel direction, building on the techniques in earlier work of Betina, Dimitrov, and Pozzi.